Developing a self-consistent AGB wind model: II. Non-classical, non-equilibrium polymer nucleation in a chemical mixture
Jels Boulangier, David Gobrecht, Leen Decin, Alex de Koter, Jeremy Yates
MMNRAS , 1–68 (2019) Preprint 27 August 2019 Compiled using MNRAS L A TEX style file v3.0
Developing a self-consistent AGB wind model:II. Non-classical, non-equilibrium polymer nucleation in achemical mixture
Jels Boulangier (cid:63) , D. Gobrecht , L. Decin , † , A. de Koter , and J. Yates Institute of Astronomy, KU Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium University of Leeds, School of Chemistry, Leeds LS2 9JT, United Kingdom Anton Pannenkoek Institute for Astronomy, Universiteit van Amsterdam, Science Park 904, NL-1098 XH Amsterdam, The Netherlands Department of Physics and Astronomy, University College London, Gower St., London WC1E 6BT, United Kingdom
Accepted 22/08/2019
ABSTRACT
Unravelling the composition and characteristics of gas and dust lost by asymptoticgiant branch (AGB) stars is important as these stars play a vital role in the chem-ical life cycle of galaxies. The general hypothesis of their mass loss mechanism is acombination of stellar pulsations and radiative pressure on dust grains. However, cur-rent models simplify dust formation, which starts as a microscopic phase transitioncalled nucleation. Various nucleation theories exist, yet all assume chemical equilib-rium, growth restricted by monomers, and commonly use macroscopic properties fora microscopic process. Such simplifications for initial dust formation can have largerepercussions on the type, amount, and formation time of dust. By abandoning equi-librium assumptions, discarding growth restrictions, and using quantum mechanicalproperties, we have constructed and investigated an improved nucleation theory inAGB wind conditions for four dust candidates, TiO , MgO, SiO and Al O . Thispaper reports the viability of these candidates as first dust precursors and reveals im-plications of simplified nucleation theories. Monomer restricted growth underpredictslarge clusters at low temperatures and overpredicts formation times. Assuming thecandidates are present, Al O is the favoured precursor due to its rapid growth atthe highest considered temperatures. However, when considering an initially atomicchemical mixture, only TiO -clusters form. Still, we believe Al O to be the primecandidate due to substantial physical evidence in presolar grains, observations of dustaround AGB stars at high temperatures, and its ability to form at high temperaturesand expect the missing link to be insufficient quantitative data of Al-reactions. Key words: stars: AGB and post-AGB – stars: winds, outflows – astrochemistry –methods: numerical
Low and intermediate mass (initially . to (cid:12) ) starsevolve through the asymptotic giant branch (AGB) phaseat the end of their life time. During this phase, AGB starslose vast amounts of material to their surroundings viaa stellar wind and thereby contribute significantly to thechemical enrichment of the interstellar medium. As low(and intermediate) mass stars dominate the initial massfunction, AGB stars are one of the main contributors of thischemical enrichment. The generally accepted hypothesis isthat the mechanism triggering the onset of the AGB stellar (cid:63) E-mail: [email protected] † E-mail: [email protected] wind is a combination of stellar pulsations and radiationpressure on newly formed dust grains (Habing & Olofsson2004). While dynamic models incorporating this scenariocan explain observed wind mass loss rates and velocities ofcarbon-rich winds (Woitke 2006a), a substantial fine-tuningis needed for oxygen-rich winds (Woitke 2006b) and a modelfrom first principles incorporating all physics and chemistrydoes not yet exist.Current AGB wind models implement dust growth byaccretion of gas onto tiny solid particles, so-called seeds,based on the prescription of Gail & Sedlmayr (1999).Such seed particles are either predicted using a nucleationtheory (e.g. Gail & Sedlmayr 1988; Helling & Winters © a r X i v : . [ a s t r o - ph . S R ] A ug J. Boulangier et al. (e.g. Ferrarotti & Gail 2006; H¨ofner et al. 2016;Dell’Agli et al. 2017). To understand the wind formationmechanism from first principles, it is essential to use anucleation theory. However, the most complex nucleationtheories still assume chemical equilibrium, restrict growthof nucleation clusters to addition of monomers, and applymacroscopic properties of solids to describe clusters of a fewmolecules. Nonetheless, progress has been made regardingthese assumptions, in a range of astrophysical fields whereunderstanding dust formation crucial (e.g. in supernovae,brown dwarf atmospheres, and the interstellar medium).First, the assumption of chemical equilibrium is discardedby e.g. Sarangi & Cherchneff (2015); Gobrecht et al. (2016);Sluder et al. (2018) who treat nucleation as consecutivechemical reactions. From a chosen cluster size, they allowdust growth by coagulation of clusters, controlled by vander Waals forces (Jacobson 2013). The chosen cluster size istypically less than 5 monomer units. As nucleation reactionrate coefficients are rarely known, these coefficients are oftenestimated and usually neglect the temperature dependenceof the reaction. The latter is crucial to infer dust formationrates as a function of the radial distance from the AGBstar. Second, the use of bulk solid properties for molecularclusters is abandoned by e.g. K¨ohler et al. (1997); Goumans& Bromley (2012); Lee et al. (2015); Bromley et al.(2016); Lee et al. (2018) by adopting chemical potentialenergies from detailed quantum mechanical calculations.When describing the clustering of gas phase molecules itis inaccurate to use extrapolated bulk properties, such asbinding energy and surface tension, firstly because clusterbinding energies are generally significantly reduced withrespect to the bulk one, and secondly because microscopicclusters do not resemble the shape/structure of the solid(Johnston 2002; Lamiel-Garcia et al. 2017; Gobrecht et al.2017). E.g., small clusters do not have well-defined surfaceslike solids, rendering the use of surface tension meaningless.Third, as far as we know, no astrophysical models existwhere the nucleation and the growth are not restricted byspecific cluster size additions (e.g. monomers or dimers). Yetpolymer and more complex nucleation theories have beendeveloped in non-astrophysics disciplines, e.g. nano andsolid-state physics. Clouet (2009, and references therein)provides a good overview of different complexity levels ofnucleation theory from a non-astrophysical perspective.Presolar grains can be identified in meteorites, inter-planetary particles, and cosmic dust by isotopic anomaliesthat cannot be explained by physical or chemical processeswithin the Solar System. The origin of the grains can betraced by isotopic ratios of atoms in the grains (Nittleret al. 1997) and point to other nucleosynthetic environmentssuch as AGB stars or supernovae (McSween & Huss 2010).Here we focus on grains with an AGB origin. Since thefirst discovery of a presolar Al O grain by Hutcheon et al.(1994), several presolar oxides have been found of whichthe majority are Al O grains (corundum) and only a feware MgAl O (spinel) (e.g. Nittler et al. 1994; Choi et al.1998; Nittler et al. 2008). Note that Al O grains are oftenreferred to as corundum, which is the thermodynamicallymost stable solid bulk form, yet Al O exists in a variety of structural forms in presolar grains (Stroud et al. 2004,2007). Subsequently, Nittler et al. (2008) identified the firstTi-oxides in presolar grains, however they did not haveany crystallographic data that would allow to determinethe structure of the grains or even conclude if they wereTiO -grains. Later, Bose et al. (2010b) claim to have founda TiO -grain. The occurrence of Ti-bearing presolar grainsis low and their rarity is often explained by the low Tiabundance in AGB stars. Additionally, presolar silicategrains (containing Si-oxides) have been found (Nguyen &Messenger 2009; Bose et al. 2010a, 2012). A more extendedsummary of discovered presolar grains can be found inthe Presolar Grain Database (Hynes & Gyngard 2009).Besides physical evidence of presolar grains, there is alsoobservational evidence for different dust precursors in AGBwinds. Notably the µ m feature, which is found in spectraof half of all AGB stars (Sloan et al. 1996; Speck et al. 2000;Sloan et al. 2003), is thought to be caused by Al O -grains(Zeidler et al. 2013; Takigawa et al. 2015; Depew et al.2006), or MgAl O (Posch et al. 1999), or by SiO orpolymerised silicates (Speck et al. 2000). Since there is noconsensus on what causes this feature, there is still a largeuncertainty on the composition of dust in AGB winds.We investigated the viability of TiO , MgO, SiO andAl O as candidates of oxygen-rich AGB dust precursorswith a revised nucleation theory. We have improved onthe current nucleation theories by abandoning equilib-rium assumptions, discarding growth restrictions, and usingquantum mechanical properties of cluster molecules. Firstly,we evolve a nucleation system kinetically, therefore it is timedependent and not in equilibrium. Secondly, the revisedtheory also allows polymer nucleation, not just interactionsvia monomers. Thirdly, quantum mechanical properties ofmolecular clusters are calculated with high accuracy densityfunctional theory. Subsequently, these are used in chemicalinteractions between the nucleation clusters instead ofusing extrapolations from bulk material. The abundancesand formation times of the largest nucleation clusters areexamined in a closed nucleating system (no interaction withother chemical species) and in a large chemical mixture.The former assumes the monomer to be a priori presentand is unable to be destroyed into smaller species. Thelatter allows chemical interactions between all speciesand starts from a purely atomic composition. To describethe chemical interactions, we used the reduced chemicalreaction network of Boulangier et al. (2019) and extendedthis with additional reactions required to chemically coupleto the nucleation candidates.Section 2 describes the chemical evolution of a closedsystem and presents the improved nucleation theory.Section 3 justifies the chosen nucleation candidates andexplains two different nucleation models. Firstly, a closednucleating model which only considers one nucleatingspecies without interaction with other chemical species.Secondly, a comprehensive nucleating model which consid-ers all nucleating species simultaneously in a large chemical https://presolar.physics.wustl.edu/presolar-grain-database MNRAS000
Low and intermediate mass (initially . to (cid:12) ) starsevolve through the asymptotic giant branch (AGB) phaseat the end of their life time. During this phase, AGB starslose vast amounts of material to their surroundings viaa stellar wind and thereby contribute significantly to thechemical enrichment of the interstellar medium. As low(and intermediate) mass stars dominate the initial massfunction, AGB stars are one of the main contributors of thischemical enrichment. The generally accepted hypothesis isthat the mechanism triggering the onset of the AGB stellar (cid:63) E-mail: [email protected] † E-mail: [email protected] wind is a combination of stellar pulsations and radiationpressure on newly formed dust grains (Habing & Olofsson2004). While dynamic models incorporating this scenariocan explain observed wind mass loss rates and velocities ofcarbon-rich winds (Woitke 2006a), a substantial fine-tuningis needed for oxygen-rich winds (Woitke 2006b) and a modelfrom first principles incorporating all physics and chemistrydoes not yet exist.Current AGB wind models implement dust growth byaccretion of gas onto tiny solid particles, so-called seeds,based on the prescription of Gail & Sedlmayr (1999).Such seed particles are either predicted using a nucleationtheory (e.g. Gail & Sedlmayr 1988; Helling & Winters © a r X i v : . [ a s t r o - ph . S R ] A ug J. Boulangier et al. (e.g. Ferrarotti & Gail 2006; H¨ofner et al. 2016;Dell’Agli et al. 2017). To understand the wind formationmechanism from first principles, it is essential to use anucleation theory. However, the most complex nucleationtheories still assume chemical equilibrium, restrict growthof nucleation clusters to addition of monomers, and applymacroscopic properties of solids to describe clusters of a fewmolecules. Nonetheless, progress has been made regardingthese assumptions, in a range of astrophysical fields whereunderstanding dust formation crucial (e.g. in supernovae,brown dwarf atmospheres, and the interstellar medium).First, the assumption of chemical equilibrium is discardedby e.g. Sarangi & Cherchneff (2015); Gobrecht et al. (2016);Sluder et al. (2018) who treat nucleation as consecutivechemical reactions. From a chosen cluster size, they allowdust growth by coagulation of clusters, controlled by vander Waals forces (Jacobson 2013). The chosen cluster size istypically less than 5 monomer units. As nucleation reactionrate coefficients are rarely known, these coefficients are oftenestimated and usually neglect the temperature dependenceof the reaction. The latter is crucial to infer dust formationrates as a function of the radial distance from the AGBstar. Second, the use of bulk solid properties for molecularclusters is abandoned by e.g. K¨ohler et al. (1997); Goumans& Bromley (2012); Lee et al. (2015); Bromley et al.(2016); Lee et al. (2018) by adopting chemical potentialenergies from detailed quantum mechanical calculations.When describing the clustering of gas phase molecules itis inaccurate to use extrapolated bulk properties, such asbinding energy and surface tension, firstly because clusterbinding energies are generally significantly reduced withrespect to the bulk one, and secondly because microscopicclusters do not resemble the shape/structure of the solid(Johnston 2002; Lamiel-Garcia et al. 2017; Gobrecht et al.2017). E.g., small clusters do not have well-defined surfaceslike solids, rendering the use of surface tension meaningless.Third, as far as we know, no astrophysical models existwhere the nucleation and the growth are not restricted byspecific cluster size additions (e.g. monomers or dimers). Yetpolymer and more complex nucleation theories have beendeveloped in non-astrophysics disciplines, e.g. nano andsolid-state physics. Clouet (2009, and references therein)provides a good overview of different complexity levels ofnucleation theory from a non-astrophysical perspective.Presolar grains can be identified in meteorites, inter-planetary particles, and cosmic dust by isotopic anomaliesthat cannot be explained by physical or chemical processeswithin the Solar System. The origin of the grains can betraced by isotopic ratios of atoms in the grains (Nittleret al. 1997) and point to other nucleosynthetic environmentssuch as AGB stars or supernovae (McSween & Huss 2010).Here we focus on grains with an AGB origin. Since thefirst discovery of a presolar Al O grain by Hutcheon et al.(1994), several presolar oxides have been found of whichthe majority are Al O grains (corundum) and only a feware MgAl O (spinel) (e.g. Nittler et al. 1994; Choi et al.1998; Nittler et al. 2008). Note that Al O grains are oftenreferred to as corundum, which is the thermodynamicallymost stable solid bulk form, yet Al O exists in a variety of structural forms in presolar grains (Stroud et al. 2004,2007). Subsequently, Nittler et al. (2008) identified the firstTi-oxides in presolar grains, however they did not haveany crystallographic data that would allow to determinethe structure of the grains or even conclude if they wereTiO -grains. Later, Bose et al. (2010b) claim to have founda TiO -grain. The occurrence of Ti-bearing presolar grainsis low and their rarity is often explained by the low Tiabundance in AGB stars. Additionally, presolar silicategrains (containing Si-oxides) have been found (Nguyen &Messenger 2009; Bose et al. 2010a, 2012). A more extendedsummary of discovered presolar grains can be found inthe Presolar Grain Database (Hynes & Gyngard 2009).Besides physical evidence of presolar grains, there is alsoobservational evidence for different dust precursors in AGBwinds. Notably the µ m feature, which is found in spectraof half of all AGB stars (Sloan et al. 1996; Speck et al. 2000;Sloan et al. 2003), is thought to be caused by Al O -grains(Zeidler et al. 2013; Takigawa et al. 2015; Depew et al.2006), or MgAl O (Posch et al. 1999), or by SiO orpolymerised silicates (Speck et al. 2000). Since there is noconsensus on what causes this feature, there is still a largeuncertainty on the composition of dust in AGB winds.We investigated the viability of TiO , MgO, SiO andAl O as candidates of oxygen-rich AGB dust precursorswith a revised nucleation theory. We have improved onthe current nucleation theories by abandoning equilib-rium assumptions, discarding growth restrictions, and usingquantum mechanical properties of cluster molecules. Firstly,we evolve a nucleation system kinetically, therefore it is timedependent and not in equilibrium. Secondly, the revisedtheory also allows polymer nucleation, not just interactionsvia monomers. Thirdly, quantum mechanical properties ofmolecular clusters are calculated with high accuracy densityfunctional theory. Subsequently, these are used in chemicalinteractions between the nucleation clusters instead ofusing extrapolations from bulk material. The abundancesand formation times of the largest nucleation clusters areexamined in a closed nucleating system (no interaction withother chemical species) and in a large chemical mixture.The former assumes the monomer to be a priori presentand is unable to be destroyed into smaller species. Thelatter allows chemical interactions between all speciesand starts from a purely atomic composition. To describethe chemical interactions, we used the reduced chemicalreaction network of Boulangier et al. (2019) and extendedthis with additional reactions required to chemically coupleto the nucleation candidates.Section 2 describes the chemical evolution of a closedsystem and presents the improved nucleation theory.Section 3 justifies the chosen nucleation candidates andexplains two different nucleation models. Firstly, a closednucleating model which only considers one nucleatingspecies without interaction with other chemical species.Secondly, a comprehensive nucleating model which consid-ers all nucleating species simultaneously in a large chemical https://presolar.physics.wustl.edu/presolar-grain-database MNRAS000 , 1–68 (2019) ucleation in AGB winds mixture. Additionally, it elaborates on the used nucleationnetworks, the construction thereof, and the details of theused quantum mechanical data. Section 4 presents theresults of the evolution of all nucleation candidates for thedifferent model setups. Section 5 focuses on the implicationsof the model results. Section 6 discusses the limitations ofthe revised nucleation, the model setups, and compares theresults to previous studies. Finally, section 7 summarisesthis work. The appendix consists of detailed descriptionof used calculations (Apps. A–C) and an overview of allquantum mechanical data sources (App. D). Additionalfigures of the model results and the used chemical networkare available as appendices E and F. This section covers the general theory of chemical reac-tions and how to evolve such a system, i.e. chemical ki-netics (Sec. 2.1), and the construction of our improvednon-classical, non-equilibrium polymer nucleation theory(Sec. 2.2).
The evolution of the composition of a system is dictatedby a set of chemical formation and destruction reactions.Mathematically, this is a set of coupled ordinary differentialequations where the change in number density of the i thspecies is given by,d n i d t = (cid:213) j ∈ F i (cid:169)(cid:173)(cid:171) k j (cid:214) r ∈ R j n r (cid:170)(cid:174)(cid:172) − (cid:213) j ∈ D i (cid:169)(cid:173)(cid:171) k j (cid:214) r ∈ R j n r (cid:170)(cid:174)(cid:172) . (1)Here, the first term, within the summation, represents therate of formation of the i th species by a single reaction j of a set of formation reactions F i . The second term isthe analogue for a set of destruction reactions D i . Eachreaction j has a set of reactants R j , where n r is thenumber density of each reactant. The rate coefficient ofthis reaction is represented by k j and has units m ( N − ) s − where N is the number of reactants involved. To solvethe chemical evolution of a system, we use the open source krome package (Grassi et al. 2014), that is developedto model chemistry and microphysics for a wide range ofastrophysical applications.In general, the rate coefficient of a two body reactionA + B C + D (2)is given by k = ∫ ∞ σ v r f ( v r ) d v r , (3)where σ is the total cross section of an A-B collision, v r isthe relative speed between A and B, and f ( v r ) is a (relative)speed distribution. The total cross section of a two-particlecollision depends on the kinetic energy of both particles and http://kromepackage.org/ their microphysical interactions. However, the reaction isoften reduced to an inelastic collision of two hard spheresdue to lack of detailed chemical information. In this case,the total cross section is the geometrical cross section ofboth spheres, σ = π ( r A + r B ) where r A and r B are the radiiof both species. The speed distribution can be representedby the Maxwell-Boltzmann relative speed distribution, thatconsiders the motion of particles in an ideal gas, f ( v r ) = (cid:18) µ π k B T (cid:19) / π v r e − µ v r k B T , (4)where µ = m A m B m A + m B is the reduced mass of the system, k B isthe Boltzmann constant, and T is the temperature of the gas.Note that when the reaction requires an activation energy E a , the integral in equation (3) should be evaluated fromthe equivalent speed v a = (cid:112) E a / µ , rather than zero. Usingthe geometrical cross section and the Maxwell-Boltzmanndistribution, equation (3) results in k = π ( r A + r B ) (cid:115) k B T πµ (cid:18) + E a k B T (cid:19) e − Eak B T . (5)In the limit where E a (cid:29) k B T this reduces to k = π ( r A + r B ) (cid:115) k B πµ E a k B T − . e − Eak B T , (6)and has the form of a modified Arrhenius’ equation, k Ar = α T β e − γ T , (7)where α, β, and γ are constants. In the limit where there isno activation energy or when E a (cid:28) k B T , the last two termsin equation (5) reduce to and the rate coefficient is givenby k = π ( r A + r B ) (cid:115) k B T πµ , (8)which also has the modified Arrhenius’ form. Here, the lastfactor denotes the average relative speed, often quoted asthermal velocity . We assume that the nucleation process is homogeneous andhomomolecular. The former states that there are no prefer-ential sites for nucleation to start, and the latter means thatnucleation happens by addition of the same molecular typeof clusters. Heteromolecular nucleation is omitted since inthis case the number of possible reactions would increaseexponentially. Additionally, nucleation occurs in a puregas-phase condition and as such no preferential nucleationsites exist. This is different compared to nucleation that canoccur on solid-state surfaces which can act as a catalyst orwhere crystal lattice defects can reduce the energy neededfor nucleation to start. This is, however, not a vector quantity and naming this a ve-locity is therefore confusing and should be avoided. The correctterminology is average relative speed.MNRAS , 1–68 (2019)
J. Boulangier et al.
In general, a nucleation/cluster growth reaction is repre-sented by,C N + C M C N+M (9)where C N and C M are clusters of size N and M , respectively.Due to a lack of reaction rate coefficients in the literature,the rate coefficient is determined via equation (8) by assum-ing an inelastic collision where the activation energy of thereaction is much smaller than k B T and is given by k + N , M = π ( r N + r M ) (cid:115) k B T πµ N , M , (10)where µ N , M is the reduced mass of the ( N , M )-system , and r N and r M are the radii of clusters of size N and M , respec-tively. Assuming that the volume scales linearly with thesize of the clusters, the radii can be written as function ofthe monomer radius r , k + N , M = π ( N / r + M / r ) (cid:115) k B T πµ N , M . (11)Note that the assumption of a spherical cluster canbe generalised to a fractal cluster with a fractal radius r f , N = N / D f r , where D f is the fractal dimension, whichequals 3 for spheres.A cluster destruction process of an ( N + M ) -sized clus-ter is represented byC N+M C N + C M , (12)The rate coefficient can be derived from the principle ofdetailed balance which states that, at equilibrium, eachelementary process is equilibrated by its reverse process.Hereby, we assume that the destruction rate is an intrinsicproperty of the cluster and does not depend on the embed-ding system (i.e. no collisional dissociation). We thereforeassume that the cluster has enough time to relax to the low-est energy configuration between its formation and sponta-neous break-up. This assumption is consistent with the factthat we describe a cluster solely by its size and minimal en-ergy configuration. With the principle of detailed balance,the destruction rate coefficient can be determined via, n eq N + M k − N , M = n eq N n eq M k + N , M k − N , M = n eq N n eq M n eq N + M k + N , M , (13)where n eq N is the equilibrium number density of the N -sizedcluster and k + N , M is the growth rate coefficient of the re-versed reaction (Eq. 10). For a system at constant pressureand temperature, the equilibrium number distribution is de-termined by minimising its Gibbs free energy (App. A). Con-sequently, the ratio of the clusters is given by n eq N n eq M n eq N + M = n tot exp (cid:18) G N + M − G M − G N k B T (cid:19) , (14) A cluster C N of specific size N denotes a molecule that existsof N -times molecule C , e.g. (SiO) is an SiO-cluster of size 2. This assumption reduces the amount of needed information, i.e.just one molecule radius instead of N radii. It does, however, alsodecrease the accuracy of the description. where G N is the Gibbs free energy of an N -sized clusterand n tot is the total number density of the gas . It is moreconvenient to use the Gibbs free energies at standard pres-sure ( P ◦ = bar = Pa = · dyne / cm ) . Here, the su-perscript ◦ refers to a quantity evaluated at this standardpressure. Using equation (A25) this ratio is given by n eq N n eq M n eq N + M = P ◦ k B T exp (cid:18) G ◦ N + M − G ◦ M − G ◦ N k B T (cid:19) . (15)Substituting this ratio into equation (13) yields a clusterdestruction rate coefficient k − N , M = k + N , M P ◦ k B T exp (cid:18) G ◦ N + M − G ◦ M − G ◦ N k B T (cid:19) . (16)Note that the standard Gibbs free energies are often givenin kJ mol − , in which case the Boltzmann constant k B inthe exponential has to be replaced with the universal gasconstant R in kJ K − mol − . This section explains the two different nucleation descrip-tions that have been used, a monomer and polymer one(Sec. 3.1). Next, it justifies the choice of nucleation candi-dates that have been considered, namely TiO , MgO, SiOand Al O (Sec. 3.2). Additionally, it describes the two dif-ferent types of chemical nucleation networks, a closed oneand a comprehensive one (Secs. 3.3 and 3.4). The closed nu-cleating network assumes the monomer to be a priori presentand is unable to be destroyed into smaller species. No as-sumptions have been made on how the monomer has beenformed or its possible existence. The comprehensive nucleat-ing network does not assume the existence of the nucleatingmonomers and starts from a purely atomic composition. The(possible) formation of the nucleating monomers and otherchemical species is determined by a large chemical reactionnetwork. Finally, this section summarises all the additionallygathered data and performed calculations prior to runningthe nucleation models (Sec. 3.6). We consider two different nucleation descriptions, polymerand monomer nucleation. The former is the most generaland uses growth and destruction of the corresponding clus-ters described by equations (11) and (16), whereas the lat-ter uses those same equation but with M = reducing itto a monomer. We make this distinction because, to ourknowledge, most homomolecular nucleation studies assumemonomer nucleation (e.g. K¨ohler et al. 1997; Lee et al. 2015;Bromley et al. 2016; Lee et al. 2018). However, the monomerassumption is only valid when the number of monomers ismuch larger than that of any other cluster. There is no quan-titative evidence to support this assumption and it turnsout to be invalid in our parameter space (Sec. 4). Sarangi Note that this is only valid in the dilute limit, i.e. the numberof clusters is small compared to the total number of particles. For higher densities this will be even less valid, e.g. browndwarfs and planetary atmospheres. MNRAS000
In general, a nucleation/cluster growth reaction is repre-sented by,C N + C M C N+M (9)where C N and C M are clusters of size N and M , respectively.Due to a lack of reaction rate coefficients in the literature,the rate coefficient is determined via equation (8) by assum-ing an inelastic collision where the activation energy of thereaction is much smaller than k B T and is given by k + N , M = π ( r N + r M ) (cid:115) k B T πµ N , M , (10)where µ N , M is the reduced mass of the ( N , M )-system , and r N and r M are the radii of clusters of size N and M , respec-tively. Assuming that the volume scales linearly with thesize of the clusters, the radii can be written as function ofthe monomer radius r , k + N , M = π ( N / r + M / r ) (cid:115) k B T πµ N , M . (11)Note that the assumption of a spherical cluster canbe generalised to a fractal cluster with a fractal radius r f , N = N / D f r , where D f is the fractal dimension, whichequals 3 for spheres.A cluster destruction process of an ( N + M ) -sized clus-ter is represented byC N+M C N + C M , (12)The rate coefficient can be derived from the principle ofdetailed balance which states that, at equilibrium, eachelementary process is equilibrated by its reverse process.Hereby, we assume that the destruction rate is an intrinsicproperty of the cluster and does not depend on the embed-ding system (i.e. no collisional dissociation). We thereforeassume that the cluster has enough time to relax to the low-est energy configuration between its formation and sponta-neous break-up. This assumption is consistent with the factthat we describe a cluster solely by its size and minimal en-ergy configuration. With the principle of detailed balance,the destruction rate coefficient can be determined via, n eq N + M k − N , M = n eq N n eq M k + N , M k − N , M = n eq N n eq M n eq N + M k + N , M , (13)where n eq N is the equilibrium number density of the N -sizedcluster and k + N , M is the growth rate coefficient of the re-versed reaction (Eq. 10). For a system at constant pressureand temperature, the equilibrium number distribution is de-termined by minimising its Gibbs free energy (App. A). Con-sequently, the ratio of the clusters is given by n eq N n eq M n eq N + M = n tot exp (cid:18) G N + M − G M − G N k B T (cid:19) , (14) A cluster C N of specific size N denotes a molecule that existsof N -times molecule C , e.g. (SiO) is an SiO-cluster of size 2. This assumption reduces the amount of needed information, i.e.just one molecule radius instead of N radii. It does, however, alsodecrease the accuracy of the description. where G N is the Gibbs free energy of an N -sized clusterand n tot is the total number density of the gas . It is moreconvenient to use the Gibbs free energies at standard pres-sure ( P ◦ = bar = Pa = · dyne / cm ) . Here, the su-perscript ◦ refers to a quantity evaluated at this standardpressure. Using equation (A25) this ratio is given by n eq N n eq M n eq N + M = P ◦ k B T exp (cid:18) G ◦ N + M − G ◦ M − G ◦ N k B T (cid:19) . (15)Substituting this ratio into equation (13) yields a clusterdestruction rate coefficient k − N , M = k + N , M P ◦ k B T exp (cid:18) G ◦ N + M − G ◦ M − G ◦ N k B T (cid:19) . (16)Note that the standard Gibbs free energies are often givenin kJ mol − , in which case the Boltzmann constant k B inthe exponential has to be replaced with the universal gasconstant R in kJ K − mol − . This section explains the two different nucleation descrip-tions that have been used, a monomer and polymer one(Sec. 3.1). Next, it justifies the choice of nucleation candi-dates that have been considered, namely TiO , MgO, SiOand Al O (Sec. 3.2). Additionally, it describes the two dif-ferent types of chemical nucleation networks, a closed oneand a comprehensive one (Secs. 3.3 and 3.4). The closed nu-cleating network assumes the monomer to be a priori presentand is unable to be destroyed into smaller species. No as-sumptions have been made on how the monomer has beenformed or its possible existence. The comprehensive nucleat-ing network does not assume the existence of the nucleatingmonomers and starts from a purely atomic composition. The(possible) formation of the nucleating monomers and otherchemical species is determined by a large chemical reactionnetwork. Finally, this section summarises all the additionallygathered data and performed calculations prior to runningthe nucleation models (Sec. 3.6). We consider two different nucleation descriptions, polymerand monomer nucleation. The former is the most generaland uses growth and destruction of the corresponding clus-ters described by equations (11) and (16), whereas the lat-ter uses those same equation but with M = reducing itto a monomer. We make this distinction because, to ourknowledge, most homomolecular nucleation studies assumemonomer nucleation (e.g. K¨ohler et al. 1997; Lee et al. 2015;Bromley et al. 2016; Lee et al. 2018). However, the monomerassumption is only valid when the number of monomers ismuch larger than that of any other cluster. There is no quan-titative evidence to support this assumption and it turnsout to be invalid in our parameter space (Sec. 4). Sarangi Note that this is only valid in the dilute limit, i.e. the numberof clusters is small compared to the total number of particles. For higher densities this will be even less valid, e.g. browndwarfs and planetary atmospheres. MNRAS000 , 1–68 (2019) ucleation in AGB winds & Cherchneff (2015); Gobrecht et al. (2016); Sluder et al.(2018), however, do allow polymer nucleation but limit it tosmall clusters ( N < ). In oxygen-rich atmospheres ( C / O < ) , carbon is pre-dominantly locked-up in CO, strongly inhibiting theformation of carbonaceous dust. Highly stable moleculesin an carbon-deficient gas such CO, N , and CN only havea solid form (ice) at temperatures well below
500 K . Alsosolid oxygen only forms at extremely cold temperatures.Hence, nucleation at high temperatures must proceed viahetero-atomic species such as composite metal oxides.Monomers with high bond energies are preferential candi-dates for first nucleation because higher energies generallyallow for easier formation and more difficult destructionat higher temperatures. Therefore, bond energies of simplemetal oxides give a hint for which molecules will play apredominant role. Considering the most abundant atomicmetals in AGB winds, SiO, TiO, and AlO are the metaloxides with the highest bond energy (Fig. 1). Even thoughthe amount of Ti is almost a factor 40 and 400 lower thanAl and Si, respectively, it can still be an important moleculedue to its high bond energy. Similarly, MgO, and FeOhave lower bond energies but the high atomic abundance ofMg and Fe can make them important nucleation candidates.Although the metal oxides hint at the engaged species, themost compelling evidence for nucleation building blockscomes from presolar grains. Considering all the presolargrains that originated from AGB stars, Al O grains are themost frequently occurring oxygen-bearing ones. (Hutcheonet al. 1994; Nittler et al. 1994; Choi et al. 1998; Nittler et al.2008) . In these grains, Al O is the basic building block(repeating formula unit) that forms the bulk grains with avariety of structural forms (Stroud et al. 2004, 2007). Therepetition of such a basic building block strengthens ourassumption of homomolecular nucleation. The second mostfrequently found grains, roughly a factor less abundant,are the ones with MgAl O as repeating formula unit(Nittler et al. 1994; Choi et al. 1998; Nittler et al. 2008).Additionally, there is some evidence for silicon and titaniumoxides in presolar grains (Nguyen & Messenger 2009; Boseet al. 2010b; Nittler et al. 2008; Bose et al. 2010a). However,as only little amount of this material is detected, it isunclear what the repeating basic building block is.Considering the occurrence in presolar grains, the atomicmetal abundance, and the bond energy of simple metaloxides, we choose Al O to be our primary nucleation can-didate. Next, we do not consider MgAl O as a candidateas this molecule consist of three different atoms, makingit more complex to characterise its molecular features. Weinclude MgO as a candidate because it (and its clusters)might play a role in the formation of MgAl O grains.Additionally, we take TiO as a nucleation candidate. Even We refer to the chemical use of metals and not the astronomicalone. Bond energy is a measure of the strength of a chemical bond. − − − − Atomic abundance n i /n H B o nd e n e r g y D ( k J / m o l e ) FeONiO MgOSiOAlOTiOCoO NaOPOCrO CaOMnO N COCN
Figure 1.
Simple molecules (mainly oxides) with high bond ener-gies at
298 K (Luo 2007) and/or a high atomic abundance providehints at which species play a dominant role in the initial dustformation in AGB winds. though there is no substantial evidence for TiO to bethe repeating formula unit in presolar grains containingtitanium oxides, it is, however, the repeating basic buildingblock in other commonly found titanium minerals onEarth (e.g. rutile and anatase). Lastly, we select SiO asa candidate. Although there is no physical evidence inpresolar grains that SiO is the repeating formula unit, itdoes have the highest bond energy of the most abundantatomic metals and it most likely will play an important rolein the formation of silicate grains. We exclude FeO from thisstudy because, so far, only one potential detection of FeO inAGB circumstellar environment has been reported (Decinet al. 2018), nor has there been proof of FeO-containingparticles in pre-solar grains. Additionally, Fe-containingnanoparticles can display various magnetic behaviours suchas ferromagnetic, antiferromagnetic, ferrimagnetic, andnonmagnetic, and are therefore challenging to characterise.A typical interstellar dust grains of radius . µ m con-tains monomer units with a typical radius of roughly . (Table D1). Hence, in order to construct a dust grainvia reaction rate equations, one needs of the order of equations. As this is computationally impossible, we limitthe maximum cluster size so the largest clusters roughlyconsist of to atoms, making it still feasible to per-form high accuracy density functional theory calculations(Sec. 3.6.2). We take the largest cluster to be (TiO ) ,(SiO) , (MgO) , and (Al O ) . Note that these clustersizes are not necessarily the threshold from which thespecies can be considered as a macroscopic, solid dust grain(Sec. 6). A closed nucleation model corresponds to the evolution of acluster system according to growth and destruction rate co-efficients (Eqs. 11 and 16) with the monomer as the smallestand the maximally considered cluster size as the largest al-lowed clusters. Such a model starts with an initial monomerabundance and follows the growth of this monomer over
MNRAS , 1–68 (2019)
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Table 1.
Initial chemical composition. This is equal to the time-averaged mass fractions in the wind for a nucleosynthetic AGBevolutionary model with an initial mass of 1 M (cid:12) and metallicityZ=0.02 of Karakas (2010). The mass fraction of Ti is that of solarabundance (Asplund et al. 2009).Element i Mass fraction X i n i / n H He . · − . · − C . · − . · − N . · − . · − O . · − . · − F . · − . · − Na . · − . · − Mg . · − . · − Al . · − . · − Si . · − . · − P . · − . · − S . · − . · − Ti . · − . · − Fe . · − . · − e – − (cid:205) Ni X i = . · − time at a fixed temperature. We construct a model grid intemperature and density that is primarily applicable to anAGB wind (but that is also valid in other environments) andevolve each model over a timescale of one year. The lattercorresponds to the longest dynamically stable period (be-tween pulsation-induced consecutive shocks), resulting in aroughly constant local temperature and density in that pe-riod. For the initial abundance of the monomer we assumeall of the available atomic metal to be locked-up in themonomer (Table 1). For the available atomic metal abun-dance we choose the same composition as Boulangier et al.(2019) who take the time-averaged elemental mass fractionsin the wind from (cid:12) and Z = . AGB evolution modelof Karakas (2010) (defined as (cid:104) X ( i )(cid:105) in Karakas & Lattanzio(2007)). For Ti we take the solar abundance because thiselement is not considered in the nucleosynthesis networks ofKarakas (2010) . A comprehensive nucleation model corresponds to the evo-lution of nucleation clusters in a large chemical network ac-cording to growth and destruction rate coefficients (Eqs. 11and 16) until a specified maximum cluster size. Such a modelstarts from the atomic composition rather than the initialmonomer abundance which is used in a closed nucleation The abundance of Ti is not affected by the slow neutron cap-ture process because of low neutron capture cross sections for ele-ments below iron, and burning temperatures are not high enoughfor higher burning processes to affect Ti. Hence, (cid:104) X ( i )(cid:105) of Ti doesnot change between birth and death of low and intermediate massstars. model (Sec. 3.3). This is a more realistic prescription asis removes the assumption of the monomer being (abun-dantly) present. Moreover, it allows for more chemical in-teraction between species and the creation of other metal-bearing molecules besides the nucleation candidate clusters.In practice, the reaction network consists of the closed nu-cleation networks of TiO , MgO, SiO, and Al O (Sec. 3.3)extended with the reduced AGB wind network of Boulangieret al. (2019). However, because their reduced network doesnot consider any Ti, Al, and only a few Mg reactions, wehave added all reactions that include these elements avail-able in the literature. Additionally, where necessary and pos-sible, we have included the reversed reaction based on theassumption of detailed balance . As with the closed nucle-ation models, we compute the same grid of models in tem-perature and density over a one year period but with aninitial atomic composition (Table 1). It is instructive to investigate the nucleation of chemicalspecies in a closed system with the assumption of an a pri-ori monomer existence to gain insight in the efficiency ofthe nucleation process different species. Such preliminarynucleation investigations can already exclude candidates asviable AGB dust precursors based on inefficient nucleationat high temperatures. This pre-selection of nucleation candi-dates leads to a considerable reduction of the computationalcost when coupling the reaction network to a hydrodynam-ical framework. Moreover, a closed nucleation investigationreduces the number of uncertainties when interpreting thenucleation process. For example, the nucleation of clustersin a large chemical network might not occur due to an in-sufficient or incorrect description of the gas-phase chemistryprior to the monomer formation rather than the nucleationprocess itself, which can be very effective. By ignoring thedisentanglement between monomer formation and the nu-cleation process, the nucleation species can be wrongly dis-carded as a good dust candidate. Additionally, the closednucleation system allows us to investigate the impact of us-ing the improved nucleation description, such as monomerversus polymer nucleation and using molecular energies com-pared to bulk energies.
This section covers the additional chemical reactions, quan-tum mechanical properties and calculations needed to con-struct valuable nucleation reaction networks. The first sec-tion describes the addition of chemical reactions and thesecond section the collection and calculation of quantum me-chanical properties of molecules and clusters necessary forcertain reversed reactions. The reversed rate coefficient depends on the difference in Gibbsfree energy of reactants and products (i.e. the Gibbs free energyof reaction). If there was insufficient data in the literature tocalculate these energy values, we did not include the reversedreaction. MNRAS000
This section covers the additional chemical reactions, quan-tum mechanical properties and calculations needed to con-struct valuable nucleation reaction networks. The first sec-tion describes the addition of chemical reactions and thesecond section the collection and calculation of quantum me-chanical properties of molecules and clusters necessary forcertain reversed reactions. The reversed rate coefficient depends on the difference in Gibbsfree energy of reactants and products (i.e. the Gibbs free energyof reaction). If there was insufficient data in the literature tocalculate these energy values, we did not include the reversedreaction. MNRAS000 , 1–68 (2019) ucleation in AGB winds In order to construct a reaction network for the comprehen-sive nucleation models, reactions from atomic Ti, Al, Si, andMg up to the corresponding nucleation monomer have to beincluded. Additionally, to increase the accuracy of chemicalinteractions, as many as possible other nucleation-relatedmetal-bearing molecules should be added to the networkwith corresponding reactions. Even though some species orreactions might not be important and could be omitted,such filtering is beyond the scope of this paper becausecomputation time is currently not an issue as we onlyperform grids of models rather than coupling it in real-timeto a hydrodynamical framework.Ti-bearing molecules are not well studied and correspond-ing reaction rate coefficients are lacking in astrochemicaldatabases. We could only find 9 reactions of which onlyone had a reversed reaction. For the remaining 8 reversedreactions we assumed detailed balance. We did, however,ignore reactions for the Ti-Cl-H system (Teyssandier &Allendorf 1998) due to the low abundance of both Cl andTi in AGB stars.Apart from the SiO-nucleation reactions, just one otherSi-reaction is added relative to Boulangier et al. (2019),whose network is mainly constructed from the astrochem-ical databases UMIST (McElroy et al. 2013) and KIDA(Wakelam et al. 2012) in which Si-bearing molecules arewell-studied. The destruction of SiO by atomic hydrogen,calculated via detailed balance, is added to the chemicalnetwork to equilibrate the forward reaction. Previously,the only incorporated SiO destruction reaction was thecollision of He + , which requires very high temperatures.Additionaly, 15 Mg-related reactions are added. Onlyfor 7 of them we added a reversed detailed balance reaction.However, due to a lack of quantum chemical data onMgO , MgO , and MgO no reversed reactions for reactionsincluding such species are added. Reactions with ionisedMg-bearing molecules can be found in the literature (Whal-ley & Plane 2010; Mart´ınez-N´u˜nez et al. 2010; Whalleyet al. 2011) but are ignored because ionisation is unlikely atthe low temperatures of our grid.In total 51 Al-related reactions and their reversed de-tailed balance reactions are added, that mostly originatefrom combustion chemistry. In order to calculate the reversed reaction rate coefficientunder the assumption of detailed balance, one needs theGibbs free energy (GFE) of all reactants and products, asa function of temperature at a specific pressure (Eq. (16)for nucleation and e.g. equations ( ) − ( ) in Grassi et al.(2014) in general). In principle, one can also use the differ-ence in Gibbs free energy of formation (GFEoF) because One only needs to determine the GFE at a single pressure tobe used in reversed rate coefficients. Often a standard pressure of = · Pa is used. the additional contribution of individual atoms cancels out(App. C). On one hand, using the GFEoF has the advantageof being calculated for numerous species and being includedin different databases, e.g. so-called NASA-polynomials (Burcat & Ruscic 2005) and NIST-JANAF Thermochem-ical Tables (Chase 1998). On the other hand, there areinconsistencies between both databases such as the samespecies having different GFEoF values. By benchmarking,Tsai et al. (2017) also came to this conclusion and assignthe discrepancies between the databases to a differentlydefined reference level that corresponds to zero energy.Another reason might be that the GFEoF values rely onexperimentally determined values of quantities at roomtemperature which can have large error bars. Moreover, thedetails of the calculations or experiments are often unclearas these have been performed decades ago and frequentlylack detailed descriptions. For consistency, we use (andstrongly encourage to use) GFE rather than GFEoF.Because the GFE is an intrinsic property of a species, itdoes not rely on any experimental value at a referencetemperature (e.g. room temperature) but can be calculatedfrom first principles with absolute zero as a reference point(App. B). In short, to calculate the GFE as a function oftemperature, one only needs the total partition functionand the electronic potential energy at zero Kelvin (Eq. B9).We calculate the GFE of all clusters of the four nucleationspecies TiO , MgO, SiO and Al O by first gathering themost recent structural information (i.e. atomic coordinates)of the lowest energy isomers, i.e. the so-called global minima(Table D1). Subsequently, using gaussian09 (Frisch et al.2013), we perform density functional theory (DFT) calcula-tions including a vibrational analysis to determine the GFE.For consistency, we always use the same functional andbasis set, namely the B3LYP functional (Becke 1993) and6-311+G* basis set. Other functionals and/or basis setsmight be more accurate for specific properties or species,yet B3LYP is well established and suitable for inorganicoxides (Cor`a 2005), and 6-311+G* is a good compromisebetween accuracy and computation time.For all non-cluster species participating in reversed re-actions, we have collected the electronic potential energieswhen available (Table D2). All energies originate from DFTcalculations by the Computational Chemistry Comparisonand Benchmark DataBase (CCCBDB, Johnson 2018).For consistency we always use results of the same functionaland basis set, namely B3LYP and 6-31+G** . We performDFT calculations for the species of which no electronicpotential energies are present in any database, using thesame DFT setup as for the nucleation clusters (Table D2). http://garfield.chem.elte.hu/Burcat/burcat.html https://janaf.nist.gov/ This basis set is spanned by 6 primitive Gaussians, includesdiffusion(+) and polarisation(*). https://cccbdb.nist.gov/ CCCBDB does not contain calculations with 6-311+G*, theone we used for the nucleation clusters. The 6-31+G** basis setis slightly smaller but also includes diffusion and polarisation, andmost closely resembles 6-311+G*MNRAS , 1–68 (2019)
J. Boulangier et al.
When possible, we have gathered partition functions of the non-cluster species participating in reversed reac-tions (Table D2). These values originate from detailedcalculations and/or experiments. If no literature partitionfunctions could be found, we have calculated them from in-ternal energy levels (rotational, vibration, and electronic )found in the CCCBDB (App. B). Note that this method isless precise due to approximations such as considering thespecies as a rigid rotor and harmonic oscillator. Again, whenno energy levels were available in the literature, we havecalculated them via a vibrational analysis as a follow-up onthe DFT calculations (Table D2). This section presents the simulation results of the two mainmodel setups, one with closed nucleation networks and onewith a comprehensive chemical nucleation network. Theclosed nucleation network setup considers four nucleationspecies, TiO , MgO, SiO and Al O . Additionally, each ofthese sub-setups will use the monomer nucleation (MN) andthe polymer nucleation (PN) approach. The comprehensivechemical nucleation network setup will encompass all fourmentioned nucleation species but only use the polymernucleation approach.Because our results include four parameters (tempera-ture, gas density, cluster number density, and time), wereduce the dimensionallity to analyse the outcome. Theanalysis of the cluster size distributions in ( T , ρ ) -space islimited to the end of the simulation, i.e. after one year.Subsequently, to infer temporal effects, we choose a bench-mark constant total mass density of · − kg m − , whichis a typical value we expect in an AGB wind (Boulangieret al. 2019, fig. 10). Note that we use the total mass densityof the gas as a parameter since this value remains constantas compared to the total number density. This section covers the evolution of four nucleation speciesTiO , MgO, SiO and Al O for a closed nucleation networksetup with both the monomer nucleation (MN) and the poly-mer nucleation (PN) description. To ensure the overview, wemainly discuss the largest clusters because they are mostinteresting to understand formation of macroscopic dustgrains. Additional figures for all clusters can be found inAppendix E1. Note that this excludes the translational part because thatdepends on the number of particles and the pressure for whichone wants to calculate the total partition function. The number of electronic energy levels is truncated to be validbelow ∼
10 000 K , which is more than sufficient for the purpose ofthis paper. (TiO ) forms when the temperature drops below thesharp threshold at to , where the low (high)temperature threshold is for the lowest (highest) densities(Fig. 2). At temperatures above , its abundance dropsorders of magnitude (Figs. E1, E10). As expected, a higherdensity leads to more collisions facilitating nucleation athigher temperatures. Between roughly
950 K and the uppertemperature threshold for both MN and PN, almost all ofthe available monomers end up in (TiO ) ( >
80 per cent).However, using MN or PN yields vastly different results atlow temperatures. In this regime, roughly below
950 K , theabundance of (TiO ) drops orders of magnitude in thecase of MN in contrast to PN, where its abundance is nearlyidentical and accounts for to per cent of the availabletitanium. The low abundance in the MN case is causedby a relatively rapidly developing lack of monomers inthis temperature range, because, by design, growth is onlyallowed by the addition of monomers. Once the bulk of thematerial is clustered in N = to chains, the monomer pop-ulation becomes depleted and further growth is quenched(Fig. E1). At our benchmark density of · − kg m − , thistypically happens in less than a day. This bottleneck doesnot occur in the case of PN since, by design, all clustersare allowed to participate in the growth process (Fig. E10).Therefore, even in the case of a lack of monomers othersmall clusters can interact and form larger clusters. Inthis low temperature regime, this occurs so efficiently thatthe small clusters ( N = to ) are completely depletedand turned into large clusters. The fact that clusters ofintermediate ( N > ) size are still present is somewhatartificial since they are only allowed to grow by additionof smaller ones due to the limitation of a maximum size of N = . As these small clusters are already depleted, theintermediate growth is quenched. In reality clusters of size N = and N = can interact to form an N = sized cluster.Using MN, the abundance of the largest moleculesconverges slowest, roughly after and
60 d for (TiO ) and (TiO ) , respectively. All smaller molecules roughlyconverge after
20 d or less (Fig. E2). Using PN, the con-vergence of (TiO ) occurs faster, already after
20 d , evenin less than for the slightly smaller clusters. All smallclusters are also formed within but continue to steadilygrow into larger ones (Fig. E11).
Unlike for TiO clusters, the conditions that determine thepresence of the largest MgO cluster differ strongly betweenthe different nucleation descriptions, being more complexin the MN case. Yet both nucleation descriptions revealthat the second largest cluster (MgO) , rather than thelargest cluster (MgO) , is the most stable and thereforemost abundant cluster (Figs. E3, E12). Hence, we mainlydiscuss (MgO) . In the MN case, between and and at the highest densities all available monomers end up This happens over the entire temperature range unless statedotherwise. MNRAS000
Unlike for TiO clusters, the conditions that determine thepresence of the largest MgO cluster differ strongly betweenthe different nucleation descriptions, being more complexin the MN case. Yet both nucleation descriptions revealthat the second largest cluster (MgO) , rather than thelargest cluster (MgO) , is the most stable and thereforemost abundant cluster (Figs. E3, E12). Hence, we mainlydiscuss (MgO) . In the MN case, between and and at the highest densities all available monomers end up This happens over the entire temperature range unless statedotherwise. MNRAS000 , 1–68 (2019) ucleation in AGB winds
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Temperature (K) − − − − − D e n s i t y ( k g m − ) (TiO ) . . . . . . . . . . N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
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Temperature (K) − − − − − D e n s i t y ( k g m − ) (TiO ) . . . . . . . . . . N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
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Temperature (K) − − − − − D e n s i t y ( k g m − ) (MgO) . . . . . . . . . . N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
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Temperature (K) − − − − − D e n s i t y ( k g m − ) (MgO) . . . . . . . . . . N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
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Temperature (K) − − − − − D e n s i t y ( k g m − ) (Al O ) . . . . . . . . . . N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
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Temperature (K) − − − − − D e n s i t y ( k g m − ) (Al O ) . . . . . . . . . . N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r ) Figure 2.
Normalised mass density (or mass fraction) w.r.t. the initially available monomers after one year of (TiO ) , (MgO) , and(Al O ) for the closed nucleation models with left monomer and right polymer nucleation description. We refrain from showing (SiO) since its abundance is zero in the entire parameter space. Note that (MgO) is the second largest cluster, but most stable and moreabundant one. Monomer nucleation under predicts the amount of large clusters at low temperature, as compared to polymer nucleation.This under prediction is due to the limitation of growth-by-monomers in the monomer nucleation description. In the most favourablenucleation conditions, more than 90 per cent of the initial monomers end up in the largest cluster. Al O -clusters are the primarycandidate for first dust precursors because (Al O ) forms at the highest temperature as compared to the other candidates. Normalisednumber densities w.r.t. the initially available monomers can easily be found by dividing the normalised mass density by the cluster size,i.e. divide by 8 in the case of (Al O ) . An overview of all clusters of all candidates can be found in Appendix E1 with an in-depthanalysis in Sections 4.1 and 5.2. in (MgO) (Fig. 2). But, within this temperature range, thisamount strongly decreases with decreasing density whereat · − kg m − just 10 per cent ends up in (MgO) andat the lowest densities this amount reduces to . per cent(Fig. E3). Note that below (MgO) clusters can alsoexist but maximally take up 1 per cent of the availablemonomers. In the PN case, (MgO) -clusters already formbelow to and above they contain over 90per cent of the available monomers (Fig. 2). Below ,they are less abundant but still encompass between to per cent of the monomers. Note that below ,there is more (MgO) than (MgO) , making the largestcluster the most stable one at low temperatures (Fig. E12).Similar to the other nucleation candidates, the lack oflarge MgO-clusters at low temperatures, below , inthe MN case is due to the construction of this nucleationdescription, that limits growth by addition of monomers.It is also interesting to note that in both nucleation casesand above , cluster sizes N = , , , and aremore abundant than their direct size-neighbours. This MNRAS , 1–68 (2019) J. Boulangier et al.
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Temperature (K) T i m e ( d ) (TiO ) N u m b e r d e n s i t y ( m − )
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Temperature (K) T i m e ( d ) (TiO ) N u m b e r d e n s i t y ( m − )
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Temperature (K) T i m e ( d ) (MgO) N u m b e r d e n s i t y ( m − )
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Temperature (K) T i m e ( h ) (MgO) N u m b e r d e n s i t y ( m − )
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Temperature (K) T i m e ( h ) (Al O ) N u m b e r d e n s i t y ( m − )
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Temperature (K) T i m e ( h ) (Al O ) N u m b e r d e n s i t y ( m − ) Figure 3.
Temporal evolution of the absolute number density of (TiO ) , (MgO) , and (Al O ) at the benchmark total gas density ρ = · − kg m − with left monomer and right polymer nucleation description. Be aware of the different time scales between species.Overall, convergence with monomer nucleation description takes slightly longer than using the polymer nucleation one. It can also yieldvastly different final abundances which is most noticeable for (MgO) . We refrain from showing (SiO) since its abundance is zero inthe entire parameter space. An overview of all clusters of all candidates can be found in Appendix E1 with an in-depth analysis in Secs.4.1 and 5.2. is a consequence of the energetic stability of these MgOcluster sizes. This phenomenon would not arise when usingextrapolated bulk properties for the clusters (i.e. classicalnucleation), but only when calculating the energy on amicroscopic level (i.e. quantum mechanically).Determining the time scale of abundance convergencefor MgO-clusters is problematic, due to the complex be-haviour in temperature-space. We give a rough convergencetime scale below and above . Below and inthe case of MN, all clusters converge in just a few hours (Fig. E5). In the case of PN, the largest clusters do convergein a few hours but smaller clusters form in less than afew hours and then gradually get destroyed again over thecourse of a few days before reaching convergence (Fig. E14).(MgO) stands out as its abundance still gradually changeson time scales of to
100 d (Fig. E13). Because theevolution above is less straightforward, we limit theanalysis to the largest most stable cluster (MgO) , andrefer the reader to figures E4, E5 and E13, E14 for moredetails on all clusters. In the case of MN, the abundance of MNRAS000
100 d (Fig. E13). Because theevolution above is less straightforward, we limit theanalysis to the largest most stable cluster (MgO) , andrefer the reader to figures E4, E5 and E13, E14 for moredetails on all clusters. In the case of MN, the abundance of MNRAS000 , 1–68 (2019) ucleation in AGB winds (MgO) converges after roughly
180 d , whereas using PNthis happens in only a few hours (Fig. 3).
In both nucleation cases, the largest SiO-clusters do not formsignificantly in our ( T , ρ ) -range (Figs. E6, E15). Between to
700 K most monomers end up in (SiO) and remain inthe monomer above this temperature. Note that sizes N = to do not form at all. Since no large clusters form in our ( T , ρ ) -range, we refrain from analysing any time dependence. O For both nucleation descriptions, the largest Al O -clustersalready form at temperatures as high as to ,depending on the total gas density (Fig. 2), i.e. in hotterregimes than any of the other nucleation candidates.Moreover, between to and to more than 90 per cent of the available monomers arelocked-up in the largest cluster (Al O ) . Between thelower limits and , (Al O ) encompasses between10 and 90 per cent of the available material for the MNdescription. Below , MN again impedes a subsequentgrowth because the monomers are depleted once smallclusters have formed, resulting in a pile-up of small clustersunable to continue to grow (Fig. E7). PN does not havethis limitation and (Al O ) contains more than 50 percent of the available monomers in the entire temperaturerange below the formation threshold. Additionally PNgrowth is so efficient that the bulk of the material growsto sizes above N = , removing all smaller clusters (Fig. E16).In both nucleation cases, the formation of (Al O ) happens so fast that it is invisible on a time scale of days(Figs. E8, E17). Refining the time sampling reveals that,in both nucleation cases, convergence of the abundance of(Al O ) already occurs after roughly to
10 h (Fig. 3). ForMN, convergence happens even faster for smaller clusters(Fig. E9). For PN, however, there is a gradual creationand destruction of the smaller clusters, on a time scale ofhours (Fig. E18). Even on a time scale of
100 d , the smallestclusters do not converge but gradually get converted tolarger ones (Fig. E17).
The equilibrium abundance ratio of two clusters withdifferent sizes w.r.t to the equilibrium abundance ratioof two other cluster sizes can be calculated via Eq. (15).Such ratio of ratios can be used to more quantitativelydiscuss if clusters distributions have reached the equilibriumcomposition. Since it is most meaningful to compare ratiosif nucleation is feasible, the ratios of two smaller clustersw.r.t. the ratio of the two largest clusters are discussed inthe favourable temperature range. The results, shown forcomparison with the equilibrium abundances, correspond tothe closed PN models for the benchmark total gas density ρ = · − kg m − at the final time step (one year). Notethat if the number density of any of the four clusters speciesis below the numerical solver accuracy of · − cm − , the ratios are not shown.The relative abundances ratios of TiO - and MgO-clusters do not reach the equilibrium ratios in the entiretemperature range (Figs. E19 and E20). At the highesttemperatures, at which the nucleation is feasible, the modelresults correspond to the equilibrium ratios. However, atlower temperatures, the clusters need more time to reachthe equilibrium ratios since the interaction probability islower. This transition is visible between to and to for the TiO - and MgO-clusters, respectively.The fact that the clusters have not yet reached equilibriumratios is also visible from the temporally changing abun-dances in Figs. (E11) and (E13). The relative abundanceratios of Al O -clusters deviate more from the equilibriumratios (Fig. E21). Due to the large variation in numberdensities of the clusters in different temperature regimes(order of magnitude), it is often impossible to compareratios of the Al O -clusters. This variation is more clearlyvisible in Fig. (E17). SiO-clusters are not discussed sincethey do not significantly form in the temperature range ofinterest. This section covers the evolution of the four nucleationspecies TiO , MgO, SiO and Al O for a comprehensivechemical nucleation network with the polymer nucleation(PN) description. To ensure the overview, we mainly dis-cuss the species that also contain the cluster metals (Ti,Mg, Si, and Al) because they are most interesting to un-derstand formation of macroscopic dust grains. In analogywith Section 4.1, only the temporal evolution of the nucle-ation clusters is presented. Additional figures for all speciesof interest can be found in Appendix E2. The formation of (TiO ) occurs at the same temperatureand density conditions as in the closed nucleation modelwith the PN approach, i.e. when the temperature drops be-low the sharp threshold at to (Fig. 4). Above thisthreshold, Ti resides in either TiO , TiO, or remains atomic,with the atomic state preferred at the highest temperatures(above or higher for higher densities). (Fig. E22) Theconvergence of (TiO ) happens within roughly
40 d , simi-lar to the closed nucleation model with PN (Fig. 4). The con-vergence of other TiO -clusters is also similar to the closedPN case (Fig. E23). All available Mg remains atomic. Neither MgO, nor theMgO-clusters, nor any Mg-bearing molecules are formed.Hence we refrain from showing the abundance figures.
The abundance evolution of all SiO-clusters, in tempera-ture, density, and time, is the same as for the closed nucle-ation PN model, i.e. the large clusters do not form in the
MNRAS , 1–68 (2019) J. Boulangier et al. considered temperature-density range and the smallest clus-ters only form at the lowest temperatures (Fig. E24). Aboveroughly
700 K , all Si is locked-up in the SiO molecule (ex-cept at the highest temperatures and lowest densities, whichis due to time constraints of the simulation). This finding issomewhat in contrast to the higher binding energies of SiOcompared to SiO (Section 6.1.2). Below
700 K , the mostabundant molecules are SiO and (SiO) . Note that in theentire ( T , ρ ) -grid, Si does not remain atomic. O Most of the Al remains atomic except for some specific ( T , ρ ) -combinations. Overall creation of Al-molecules is upto maximally 1 per cent of the total available Al, exceptat the lowest temperatures for both extremes in the con-sidered density range where it can be up to roughly 50 percent (Fig. 5). The most abundant molecules are AlO, AlH,Al(OH) , and Al(OH) . Their formation regimes can be re-covered in the abundance figure of Al, and only AlO formsin the entire temperature range. Note that the figures ofless abundant Al-bearing molecules are only shown in Ap-pendix E2 since their abundance never exceeds the chosenthreshold (Fig. E25). This section interprets the nucleation model results andwhat they implicate for AGB dust precursors. Be aware thatconclusions drawn from closed nucleation networks are basedon the underlying assumption that the monomer exists andthat all of the nucleation-related metal is turned into themonomer. The reader should be cautious when using theseresults as they are not necessarily physical. They are, how-ever, useful in their own right to investigate the efficiency ofindividual nucleation species and the improved nucleationdescription.
The most prominent result is that large Al O -clusterscan form fast ( < ) at high temperatures (around to ). This makes Al O the favoured candidate tobecome the first dust particles in the inner AGB wind. Thesecond favoured candidates are MgO-clusters, which canform fast ( < ) around . We find, that (MgO) formsmore easily than the largest considered cluster (MgO) thanks to its higher stability. This is a consequence of theused non-classical nucleation description that relies on theGibbs free energy of the clusters, which is lower for (MgO) than for (MgO) , making the former more energeticallystable. Another consequence of the non-classical descriptionis the preferred cluster sizes N = , , , and , a situationthat would never occur when using a classical nucleationtheory (also noted by K¨ohler et al. 1997). The third pre-ferred dust candidates are TiO -clusters, which only formbelow at a relatively slow rate (time scale of tensof days compared to hours for MgO- and Al O -clusters).Finally, we discard SiO-clusters to be important as firstdust species as their growth requires conditions that are toocold and too dense compared to the conditions in an inner
500 1000 1500 2000 2500
Temperature (K) − − − − − D e n s i t y ( k g m − ) (TiO ) . . . . . . . . . . N o r m a li s e d m e t a l m a ss d e n s i t y ( w . r . t . i n i t i a l a t o m i c m e t a l )
500 1000 1500 2000 2500
Temperature (K) T i m e ( d ) (TiO ) N u m b e r d e n s i t y ( m − ) Figure 4.
Normalised mass density after one year (top) and tem-poral evolution of the absolute number density at the benchmarktotal gas density ρ = · − kg m − (bottom) of (TiO ) for thecomprehensive chemical nucleation models using the polymer nu-cleation description. The results are similar to the closed nucle-ation model (Fig. 2) where (TiO ) forms from to and converges within roughly
20 d . The largest cluster encom-passes more than 90 per cent of the available Ti, in the mostfavourable nucleation conditions. This implies that all atomicTi quickly forms TiO which subsequently starts to nucleate, infavourable conditions. An overview of all Ti-bearing molecules canbe found in Appendix E2 with an in-depth analysis in Sections4.2 and 5.2. AGB wind. SiO-clusters might form dust grains further outin the wind, where the temperature is below
500 K .Using the monomer or polymer nucleation descriptioncan result in substantial differences in typical formationtimes of the nucleation products, hence in their abundancesafter one year (Figs. 2, 3). The most striking difference is theabsence of large clusters at low temperatures when using theMN description. This can have profound implications whilethe wind is cooling down, underestimating the total numberof large clusters. Using the abundance of the largest clustersas a gauge of dust formation, the MN description will yieldless dust, which can delay or even hamper wind-driving.The formation time of large clusters can be several timeslarger when using the MN description. E.g. the convergenceof (TiO ) takes
60 d as compared to less than
20 d when
MNRAS000
MNRAS000 , 1–68 (2019) ucleation in AGB winds
500 1000 1500 2000 2500
Temperature (K) − − − − − D e n s i t y ( k g m − ) Al . . . . . . N o r m a li s e d m e t a l m a ss d e n s i t y ( w . r . t . i n i t i a l a t o m i c m e t a l )
500 1000 1500 2000 2500
Temperature (K) − − − − − D e n s i t y ( k g m − ) AlO − − − − N o r m a li s e d m e t a l m a ss d e n s i t y ( w . r . t . i n i t i a l a t o m i c m e t a l )
500 1000 1500 2000 2500
Temperature (K) − − − − − D e n s i t y ( k g m − ) AlH − − − − N o r m a li s e d m e t a l m a ss d e n s i t y ( w . r . t . i n i t i a l a t o m i c m e t a l )
500 1000 1500 2000 2500
Temperature (K) − − − − − D e n s i t y ( k g m − ) Al(OH) − − − − N o r m a li s e d m e t a l m a ss d e n s i t y ( w . r . t . i n i t i a l a t o m i c m e t a l )
500 1000 1500 2000 2500
Temperature (K) − − − − − D e n s i t y ( k g m − ) Al(OH) − − − − N o r m a li s e d m e t a l m a ss d e n s i t y ( w . r . t . i n i t i a l a t o m i c m e t a l ) Figure 5.
Normalised mass density after one year of the most abundant Al-bearing molecules for the comprehensive chemical nucleationmodels using the polymer nucleation description. Most Al remains atomic with up to 1 per cent in Al-bearing molecules. Al O , nor itsprecursors Al O , AlO are able to form anywhere in the considered ( T , ρ ) -grid. Hence, no Al O -clusters can form either. We believe thisissue is due to incomplete rate coefficients of Al-molecule formation reactions. An overview of all Al-bearing molecules plus a temporalevolution of Al and AlO can be found in Appendix E2 with an in-depth analysis in Sections 4.2 and 5.2. using the PN description. For (MgO) the difference is
180 d compared to a few hours (Fig. 3). Additionally, at ourbenchmark density of − kg m − the abundance of (MgO) converges to roughly m − in mere hours in the polymercase whereas in the monomer case it takes almost
200 d toconverges to only m − (Fig. 3).Although the abundance of some clusters converges,this does not happen for all clusters over the entire temper-ature regime. This result implies that no all clusters havereached equilibrium abundances yet. Hence, the assumptionof a steady state nucleation is generally not valid in theentire temperature range. Therefore, it is necessary to usea time dependent nucleation description to accurately tracethe nucleation process. Although Al O -clusters are the primary dust precur-sor candidate according to the closed nucleation models(Sec 5.1), no Al O -clusters form in the comprehensivenucleation models since the smallest building block, themonomer, cannot be created. Most Al remains atomic,though up to maximally 1 per cent can form molecules(AlO, AlH, Al(OH) , and Al(OH) , Fig. 5). The secondfavoured candidates, MgO-clusters, do not exist eitherbecause all the available Mg remains atomic. The thirdfavoured candidates according to the closed nucleationmodel, TiO -clusters, form equally efficient in the compre-hensive nucleation model. Lastly, as in the closed nucleationmodels, SiO-clusters are discarded as first dust precursorsin the considered temperature-density regime.These results suggest that, of the considered candi- dates, TiO -clusters are the only possible dust precursors.However, firstly there is ample evidence that pre-solarAGB grains mainly encompass Al O -grains rather thanTiO -grains (Hutcheon et al. 1994; Nittler et al. 1994;Choi et al. 1998; Nittler et al. 2008; Bose et al. 2010b).Secondly, dust has been observed to exist close to AGBstars, at ∼ . R (cid:63) for R Dor (Khouri et al. 2016), at < R (cid:63) for R Dor, W Hya and R Leo (Norris et al. 2012), and at < R (cid:63) for W Hya (Zhao-Geisler et al. 2015; Ohnaka et al.2016). The temperature corresponding to those spatialregions is roughly to , which is higher than theformation temperature of (TiO ) , that is around to (Figs. 2, 4). Large MgO and Al O -clusters, however,are able to form at such high temperatures (Fig. 2). Bothobservational arguments question the viability of TiO -clusters as first dust species and favour Al O -clusters,yet our comprehensive model does not predict this. Thisdiscrepancy indicates that our current model lacks chemicalreaction physics to form Al O monomers. Since we cannotform any of the two Al O precursors either (Al O andAlO , Tab. 2), we believe that the current reaction ratecoefficients involving Al-oxides are incorrect and needrevision or that alternative small Al O -cluster formationpathways are missing. This section discusses the limitations of our models (Sec. 6.1)and compares our model results with other literature studies(Sec. 6.2).
MNRAS , 1–68 (2019) J. Boulangier et al. a AlO + AlO + M Al O + M a Al + AlO + M Al O + M a Al O + O + M Al O + M b AlO + O + M AlO + M b AlO + O AlO + O a AlO + CO AlO + CO a Al O + O + M Al O + M a AlO + AlO + M Al O + MRate coefficients are determined by: a) Reversed of Catoire et al.(2003); Washburn et al. (2008) via detailed balance. b) Sharipovet al. (2012). Table 2.
Formation of Al O can only occur via Al O or AlO ,according to the reactions available in the literature. M is byconvention a third body which can be any chemical species. This section focuses on the limitations of the improvednucleation theory (Sec 6.1.1), the used chemical reactions(Sec 6.1.2), and the inference of dust properties (Sec 6.1.3).
Our non-classical, non-equilibrium nucleation theory hassome limitations. The most prominent one is most likelythat it describes the growth of clusters as an inelasticcollision between rigid spheres. This assumption does notaccount for the shape of the clusters nor mutual interactionforces. Using detailed chemical reaction coefficients for eachcluster reaction, which account for possible energy barriers,would be a large improvement. Unfortunately, such informa-tion does not yet exist. Recently, Sharipov & Loukhovitski(2018) have calculated rate coefficients of the dimerisation ofAl O based on Rice-Ramsperger-Kassel-Marcus (RRKM)theory, which is a more realistic apprximation than theusing the rigid spheres. We show both approximations asan example on how much the coefficients can differ (Fig. 6).Similarly, Suh et al. (2001) and Bromley et al. (2016) havedetermined SiO-clustering rate coefficients with RRMK the-ory. Additionally, in the cluster growth coefficient (Eq. 11),we write the radius of each cluster as a function of themonomer radius. However, since we know the shape of eachcluster, it is possible to calculate an effective radius for eachcluster, yielding a more correct geometrical cross-sectionbetween cluster collisions. Another limitation is set byusing spontaneous clusters destruction reactions that relyon detailed balance. Incorporating chemical or collisionallyinduced destruction reactions would increase the accuracyof the model. Furthermore, the entire nucleation processis assumed to be homomolecular. There is, however, nogood reason that it cannot be heteromolecular. Hetero-molecular nucleation is most likely necessary to createMgAl O -clusters, which are abundant in pre-solar AGBgrains, or Mg-containing silicates (Goumans & Bromley2012). Including heteromolecular nucleation will increasethe number of possible reactions exponentially and willincreases the amount of detailed quantum mechanical cal-culation and data needed for those reaction rate coefficients.
500 1000 1500 2000 2500 3000
T (K) − − − k ( c m s − ) RigidSpheresLindemann
Figure 6.
The reaction rate coefficients of Al O + Al O (Al O ) with the approximation of a collision of rigid spheres,used in this work, and calculated with Rice-Ramsperger-Kassel-Marcus theory plus a Lindemann fit by Sharipov & Loukhovitski(2018). As this latter also depends on the total number density,we have chosen a typical value for the inner AGB wind, n tot = m − . Our approximation over predicts the dimerisation byroughly an order of magnitude compared to the more realisticcoefficient using the Lindemann fit. Therefore, using a rigid sphereapproximation, as in this work, will most likely overestimate theefficiency of the nucleation process.
The assumption that nucleation starts with the formationof the monomer is not yet established. Small clustersmight be formed via pathways which bypass the monomermolecule. This could possibly solve the issue of not formingAl O -monomer in our models. Additionally, the fact thatnucleation occurs via the addition of monomer-multipleswith a fixed stoichiometry is not established either. Clusterscould possibly grow via the addition of other stoichiometricratios, as investigated by Patzer et al. (2005) for smallaluminium oxide clusters.Note that the used nucleation description considers theprocess as a statistical ensemble of particles which all havethe same mean temperature. However, as this is a processof molecular interactions, the notion of ‘temperature’ canbecome unclear. In reality, the particles have a temperaturedistribution around a mean kinetic temperature. Moleculardynamics simulations, which do not rely on a mean temper-ature, reveal that small temperature fluctuations amongstparticles initiate the nucleation process (Tanaka et al. 2011;Diemand et al. 2013; Toxvaerd 2015).A last limitation is the artificial maximum cluster size. Inreality, the clusters would continue to grow to form solidmaterial. This material can then, on its turn, sublimateand return nucleation species to the gas phase. Whetherthe sublimation process returns small clusters, monomers,atoms, or simple molecules is unclear. Additionally, to esti-mate the sublimation rate one needs the binding energies ofthe surface layer of the solid material. However, the phasetransition process to a solid dust grain is often describedby one fast reaction (e.g. Huang et al. 2009; Bojko et al. MNRAS000
The assumption that nucleation starts with the formationof the monomer is not yet established. Small clustersmight be formed via pathways which bypass the monomermolecule. This could possibly solve the issue of not formingAl O -monomer in our models. Additionally, the fact thatnucleation occurs via the addition of monomer-multipleswith a fixed stoichiometry is not established either. Clusterscould possibly grow via the addition of other stoichiometricratios, as investigated by Patzer et al. (2005) for smallaluminium oxide clusters.Note that the used nucleation description considers theprocess as a statistical ensemble of particles which all havethe same mean temperature. However, as this is a processof molecular interactions, the notion of ‘temperature’ canbecome unclear. In reality, the particles have a temperaturedistribution around a mean kinetic temperature. Moleculardynamics simulations, which do not rely on a mean temper-ature, reveal that small temperature fluctuations amongstparticles initiate the nucleation process (Tanaka et al. 2011;Diemand et al. 2013; Toxvaerd 2015).A last limitation is the artificial maximum cluster size. Inreality, the clusters would continue to grow to form solidmaterial. This material can then, on its turn, sublimateand return nucleation species to the gas phase. Whetherthe sublimation process returns small clusters, monomers,atoms, or simple molecules is unclear. Additionally, to esti-mate the sublimation rate one needs the binding energies ofthe surface layer of the solid material. However, the phasetransition process to a solid dust grain is often describedby one fast reaction (e.g. Huang et al. 2009; Bojko et al. MNRAS000 , 1–68 (2019) ucleation in AGB winds To infer abundances of the largest nucleation clusters, itis crucial to correctly predict the creation of its funda-mental building block, the monomer. Hence, the chemicalreaction path ways from atoms to monomers have tobe accurate. However, astrochemical databases lack thenecessary monomer formation reactions. Yet, there areindividual studies that provide some reactions, but theyare scarce depending on the nucleation candidate. Todetermine the AGB dust precursors, we believe that Ti andAl reactions are the most pressing. There are hardly anyTi-related reactions (App. F) and most Al-related reactionshave extremely high reaction barriers. The latter mainlyoriginate from combustion studies and are therefore oftenonly determined in the high density limit. Moreover, mostAl-related reaction rate coefficients are determined fromdestruction of larger molecules, which is the opposite ofwhat is actually needed. Therefore, the growth coefficientsrely on the assumption of detailed balance.It is important that the entire chemical network con-tains sufficient reactions with accurate rates. As pointedout by Boulangier et al. (2019), we are largely dependenton the astrochemical databases which do not contain all thereactions that are necessary. Due to the lack of reactionsrate coefficients and especially the unknown temperaturedependence, caution is advised when interpreting chemicalevolution results and the existence of certain moleculesbased on the gas temperature.
This work focuses on nucleation clusters to infer AGB dustproperties such as abundance, composition and formationtimes. However, the largest clusters that we consider are onlya fraction of the size of a dust grain nor do they resemblethe bulk geometry. The largest clusters’ radii range from . to .
71 nm whereas dust grains can be as large as afew micron. Lamiel-Garcia et al. (2017) predict that TiO -clusters only resemble the bulk geometry from N ≥ .For highly ionically bonded materials such as MgO-clustersthis can already be at N = due to the strong electrostaticinteractions between atoms. Therefore, one has to be carefulwhen using nucleation clusters as a gauge for dust grains.Yet, due to computational constraints a small cross-over size,from clusters to dust, has to be chosen. From this cross-over size, the particles should not be considered as molecularclusters any more but as tiny grains which can numericallybe binned in size and can grow via various physical processes(e.g. Jacobson 2013; Grassi et al. 2017; McKinnon et al.2018; Sluder et al. 2018). Because of our artificial maximumcluster-size, one has to be cautious when interpreting theabundances of the largest clusters in this work since in realitythese will most likely continue to grow to actual dust grains. This section compares our model results with other nucle-ation models (Sec. 6.2.1), with seed particle requirementsof dynamical wind models (Sec. 6.2.2), and with molecularobservations of AGB stars (Sec. 6.2.3).
In contrast to Gobrecht et al. (2016), our most completemodel (comprehensive network with polymer nucleation)does not produce any Al O -clusters. However, unlike thiswork, Gobrecht et al. (2016) used a simplified formulation todetermine reversed formation rates for Al-bearing moleculesresulting in a temperature independent rate coefficients.Some key formation reactions reveal that the used ratecoefficients can differ by up to 10 orders of magnitude (e.g.AlO + AlO + M Al O + M, Fig. 7. Note that Sluderet al. (2018) use an even higher rate coefficient for thisreaction.). Such large differences could explain differentresults of Gobrecht et al. (2016), as compared to this work.Moreover, we give a more realistic rate description byincorporating a temperature dependence in addition to thestrong density dependence which is crucial to investigatethe existence of large clusters and dust grain as a functionof temperature (e.g. Al O + O + M Al O + M,Fig. 7). Compared to observations, Gobrecht et al. (2016)overpredict the abundance of Al-bearing molecules (AlOand AlOH), whereas our models agree better with the mostrecent observations (Sec. 6.2.3).An approach similar to this work has recently beenused by Savel’ev & Starik (2018) who investigated thenucleation of Al O -clusters up to a cluster size of 75during the combustion of aluminized fuels. Similarly, theyalso model the nucleation kinetically with a set a chemicalreactions. Their nucleation reactions, however, only con-sider monomer interactions. They do consider much largerclusters than we do. However, the authors rely on estimates(interpolations) of the Gibbs free energies for N = − and do not perform DFT calculations of the global minimacandidates. The authors do not provide the geometriesof these larger sized Al O -clusters. Therefore, we cannotverify these isomers with the lowest-energy structures usedin the present study. Moreover, it is difficult to compareresults since their environment has a density of severalorders of magnitude higher making the nucleation occur onmilli- and microsecond time scales. See Starik et al. (2015)for a recent review of modelling aluminium nanoparticles inthe fuel combustion community.The nucleation efficiency of species is often determined bythe steady state nucleation rate, J ∗ / n H , which representsthe number of dust seed particles formed per second pertotal number of hydrogen. However, this rate relies on twomain assumptions. Firstly, growth of clusters only occursvia addition of monomers. Secondly, the system of clustersis in a steady state, i.e. the number densities of all clustersremain constant over time, ergo chemical equilibrium.This latter implies that the net formation of all clustersis the same and size independent. Detailed derivations forthe steady state nucleation rate can be found in Patzer MNRAS , 1–68 (2019) J. Boulangier et al. et al. (1998) but the notation used by Bromley et al.(2016) is clearer. The latter explicitly shows that J ∗ / n H solely depends on the amount of monomers and all ratecoefficients between clusters. To determine this equilibriumabundance, one has to know the Gibbs free energy of thelowest energy configuration for all cluster sizes (App. A).This data is unavailable for large clusters. It is often unclearhow this abundance is determined in nucleation papers,either from the vapour pressure of the monomer and thesolid form (as explained by Patzer et al. 1998; Helling &Woitke 2006) or by chemical equilibrium calculationsof the gas without considering the clusters (e.g. Jeonget al. 2003; Lee et al. 2015) . Because the steady statemonomer nucleation description differs significantly fromours and requires knowing the equilibrium abundance of themonomer, it is difficult to compare with. Using J ∗ / n H , it isoften claimed that only TiO nucleates efficiently enoughto form the first dust precursor. We limit the comparison toour monomer nucleation description since the steady stateone also assumes nucleation by monomers. When comparingour results with Jeong et al. (2003, fig. 1), we note that bothpredictions of (TiO ) -clusters have a steep cut-off around (Fig. 2). However, our time dependent descriptiondoes not yield the high nucleation that the steady state onedoes at low temperatures since the availability of monomersdecreases quickly hereby quenching the growth process.Additionally, the assumption of steady state is invalidsince there is a clear time dependence in cluster growth(Fig. 3). Jeong et al. (2003) exclude Al O -clusters to be aprimary dust precursor due to the low J ∗ / n H . One shouldbe careful with interpreting this result since, as they pointout, this is due to the low equilibrium abundance of themonomer and not necessarily the capability of nucleatingAl O -clusters. They do not discuss the efficiency of Al O vs TiO -nucleation based on stability of the clusters. Wefind that, if Al O -monomers could exist, they will nucleateat much higher temperatures than TiO (Fig. 2). However,we are also unable to form the Al O -monomers with aninitial atomic gas (Sec. 4.2.4).Our results indicate that Al O -nucleation is dominant athigh temperatures but the formation of the monomer viachemical reactions is unattainable with currently availabledata. Moreover, there is experimental evidence that smallAl O -clusters do exist when vaporising the solid material Since a steady state is assumed, this refers to the number ofmonomers at chemical equilibrium. A detail which is usually over-looked. Determining the equilibrium monomer abundance from thephase equilibrium with the bulk material via the vaporisationpressure inherently assumes that the bulk material exists. How-ever, since we are investigating the existence of bulk material canactually happen in certain conditions, such assumption shouldnot be made. Though we could not confirm which of these two methodsJeong et al. (2003) used, we note that if the vapour pressure wasused than the nucleation of Al O should be higher than that ofTiO since the former has a lower vapour pressure. According tothat method, this means less monomers thus all material is in thesolid form. However, they find a smaller J ∗ / n H for Al O than forTiO .
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Gas temperature (K) − − − − R a t ec o e ffi c i e n t ( c m s − ) Gobrecht et al. 2016This workAl + O + M → AlO + MAlO + AlO + M → Al O + MAl O + O + M → Al O + M Figure 7.
The reaction rate coefficients of some key Al O for-mation reaction used by Gobrecht et al. (2016) are to ordersof magnitude higher than the ones used is this work. These largedifferences could explain why Gobrecht et al. (2016) form Al O and we do not. Moreover, they estimate the barrierless three-bodyreactions of the type A + B + M AB + M to be tempera-ture independent, hampering an investigating of the temperaturedependence for Al O -cluster formation. (van Heijnsbergen et al. 2003; Demyk et al. 2004; Sierkaet al. 2007) This is a clear incentive for the scientificcommunity to investigate rate coefficients of Al-bearingreactions at high temperatures. Without this data, it willremain unclear which species forms the first dust precursorsin AGB winds. H¨ofner et al. (2016) show that the minimal normalisednumber of Al O dust seed particles (assumed to be clustersof size N = ) for driving an AGB wind is of the orderof n s / n H ∼ − , with n s the seed particle number density.For comparison, we do a rough extrapolation of our resultsby assuming that all the largest Al O -clusters get turnedinto clusters of size N = . This is in line with rapid for-mation of the largest clusters and depletion of the smallestones (Sec. 4.1.4). Since our largest cluster has roughly size N = , the number of seed particles of N = wouldbe 100 times smaller. If we compare with Al O -clustersand assume that 1 per cent of the available Al turns intoAl O (Sec 4.2.4 and 5.2), then the total number of largestclusters is roughly 10 per cent of the initial number ofmonomers. This translates to n (Al O ) / n Al ≈ − .Using n Al / n H from Table 1, this yields a normalisednumber of seed particles n (Al O ) / n H ≈ · − , whichis already
100 000 times more than needed according tothe models of H¨ofner et al. (2016). We can also comparethis with the number of (TiO ) -clusters. Here no as-sumption on the number of monomers has to be madebecause the comprehensive network model with PN alreadypredicts the amount of (TiO ) . This is roughly 10 percent of the available number of Ti. Again assuming thatthe number of (TiO ) -clusters is 100 times smallerthan (TiO ) and using the initial n Ti / n H from Table 1, MNRAS000
100 000 times more than needed according tothe models of H¨ofner et al. (2016). We can also comparethis with the number of (TiO ) -clusters. Here no as-sumption on the number of monomers has to be madebecause the comprehensive network model with PN alreadypredicts the amount of (TiO ) . This is roughly 10 percent of the available number of Ti. Again assuming thatthe number of (TiO ) -clusters is 100 times smallerthan (TiO ) and using the initial n Ti / n H from Table 1, MNRAS000 , 1–68 (2019) ucleation in AGB winds yields n (TiO ) / n H ≈ · − . This is in line with the(Al O ) -cluster extrapolation. Our prediction of TiO -clusters (Fig. 4) agrees withKami´nski et al. (2017) who state that there is no solid TiO close to the star ( T > K). They also claim that TiOand TiO are abundantly present in the extended envelope( to
500 K ) and therefore TiO -clusters should not sig-nificantly exist to aid in wind driving. However, accordingto models of H¨ofner et al. (2016), a tiny faction of seedparticles ( n s / n H ∼ − ) can be sufficient to aid in winddriving (Sec. 6.2.2). The lower left corner of our ( T , ρ ) -gridmost closely resembles the extended envelope regime (i.e.cold and sparse), which shows that the TiO molecule andTiO -clusters can simultaneously be present (Fig. E22).When intuitively extrapolating to lower temperatures andlower densities, as if moving further out into the extendedenvelope, we expect a higher TiO and TiO abundance andless TiO -clusters.Khouri et al. (2018) observe that for the oxygen-richAGB star o Cet 4.5 per cent of the atomic Ti is locked-up inTiO . It is however difficult to compare with our model gridsince the presence of the molecule is extremely sensitive togas temperature and its abundance ranges from to per cent of the intitial atomic Ti (Fig. E22). As it is unclearwhat the temperature coverage of the observation is, thederived abundance is most likely an average in a certaintemperature range. Kami´nski et al. (2016) discovered AlO,AlOH, and AlH in o Cet but could only determine theabundance of AlO. They find n AlO / n H = − − − , whichagrees with our model predictions that maximally 1 percent of all Al is turned into molecules, with AlO the mostabundant molecule ∼ n AlO / n H < − . Kami´nski et al.(2016) do state that AlOH is present in a gas temperatureof ±
170 K , and that AlH is detected between . to (cid:63) . Both observational constraints comply with our modelpredictions (Fig. E25). Additionally, Decin et al. (2017)find that for AGB stars IK Tau and R Dor the amount ofAlO, AlOH, and AlCl accounts for maximally 2 per centof the total aluminium budget. Both observations are inline with our prediction that maximally 1 per cent of all Alis turned into molecules (Fig. 5). The amount of detectedAlOH in R Dor only accounts for roughly 0.02 per cent,yet this is still significantly more than our models predict(Fig. E25). Lastly, Khouri et al. (2018) also deduce that lessthan 0.1 per cent of the atomic Al is converted into AlO.In conclusion, all three observational studies agree with ourprediction that maximally 1 per cent of all Al is turned intomolecules. Our prediction also better supports the recentobservations than the significantly higher abundances ofAl-bearing molecules predicted by models of Gobrecht et al.(2016).As both observations and our comprehensive modelagree that maximally 1 per cent of all atomic Al turnsinto a molecule (Sec. 4.2.4), it is interesting to analyse theresults of a closed nucleation model with only 1 per cent of
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (Al O ) . . . . . . . . . . N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) (Al O ) N u m b e r d e n s i t y ( m − ) Figure 8.
Normalised mass density after one year (top) and tem-poral evolution of the absolute number density at the benchmarktotal gas density ρ = · − kg m − (bottom) of (Al O ) for theclosed nucleation model with an initial Al O abundance of 1 percent of the available Al using the polymer nucleation description.The results are similar to the closed nucleation model with all Alturned into Al O (Figs. 2, 3). Due to the lower amount of speciesthe formation threshold is slightly lower at to andconvergence takes a little longer, roughly
20 d . the available Al as initial Al O abundance. We choose toonly use the polymer nucleation description. Compared to a100 per cent initial abundance, the temperature formationthreshold of (Al O ) has slightly lowered to to (Fig. 8). This is expected as a lower density produces lesscollisions therefore making it more difficult to form clustersat higher temperatures. Similarly, (Al O ) converges onlyafter roughly
20 d which is significantly longer than the to
10 h for the 100 per cent initial abundance model (Fig. 8).Besides the temporal effects, the results are analogous tothe 100 per cent case where (Al O ) also contains morethan 90 per cent of the available monomers at the highestformation temperatures. In this paper, we have constructed and investigated animproved nucleation theory by abandoning the assumptionof chemical equilibrium, dropping the restriction of cluster
MNRAS , 1–68 (2019) J. Boulangier et al. growth by only monomers, and using accurate quantummechanical properties of molecular clusters. We haveexamined the viability of TiO , MgO, SiO and Al O as candidates of the first dust precursors in oxygen-richAGB winds. The choice of candidates is based on rigoroustheoretical and observational evidence (Sec. 3.2).This work consists of two main nucleation descriptions, onethat only allows cluster growth via monomers and one thatallows polymer interaction. Both assume the nucleationprocesses to be homogeneous and homomolecular. Withthese descriptions, two main types of systems are evolvedin a grid of temperature and density that is typical forAGB winds: a closed nucleation system and a compre-hensive chemical nucleation system. The former considersthe growth of one nucleation candidate species with themonomer as the smallest building block and assumes thatall available atomic metal is locked-up in the monomer.The latter allows chemical interaction between species ina gas mixture which includes all nucleation species andstarts with an atomic composition. The former providesinsight in the nucleating efficiency of each candidate intemperature and density space, and the latter yields amore complete chemical nucleation model by removing theassumption of the a priori existence of the monomer (Sec. 3).Constructing the nucleation reaction networks requiredquantum mechanical data of all clusters, which we calcu-lated with high precision density functional theory. Sincesuch calculations exponentially increase with cluster size,we limit the maximal size to roughly N = . The compre-hensive chemical reaction network is constructed by addingrelevant chemical reactions to an already carefully designedreduced network for AGB winds (Boulangier et al. 2019).The extension includes all relevant and available reactionsto form the nucleation monomers. Since a significantamount of reversed reactions is not present in the literature,quantum mechanical data for the participating species isneeded to calculate those reaction rate coefficients. We havegathered as much as possible data from the literature andperformed density functional theory calculations when thiswas unavailable (Sec. 3.6).Overall, using the monomer nucleation description ascompared to the polymer one, will underestimate theabundance and overestimate the formation time of thelarge clusters. Using the abundance of the largest clustersas a gauge of dust formation, the monomer nucleationscenario would underestimate the amount of dust and over-estimate its formation time. This can lead to less efficientwind-driving or even the absence of a wind in theoreticalsimulations. The monomer description also inhibits theformation of large clusters at low temperatures due to arapidly developing lack of monomers, which by design is theonly growth mechanism. The polymer description does notsuffer from this limitation and is therefore more realistic.Comparison with equilibrium abundance ratios revealsthat the assumption of equilibrium is not valid over theentire temperature range for a period of one year. Hence, atime-dependent description in necessary to investigate thenucleation process in AGB winds. The closed nucleation models, which assume that thenucleation monomers are present, predict that Al O isthe primary candidate to be the first AGB dust precursor.These clusters rapidly form at much higher temperaturesthan any other cluster, around to and in lessthan a few days. Rapid dust formation at high temperatureswill aid in driving the AGB wind, since the wind is coolingdown from hot shocks ( ∼
10 000 K , Boulangier et al. 2019).At around to , large MgO-clusters can form andonly at to large TiO -clusters arise. Formationof SiO-clusters is not favourable in the considered temper-ature range but requires colder conditions. Note that theabove conclusions are drawn on the underlying assumptionthat the monomer exists (Sec. 4.1).The comprehensive chemical nucleation model yieldsdifferent results from the closed nucleation ones. Firstly,it does not predict any Al O -clusters, nor its monomer,nor its molecular precursors (Al O and AlO ) but mostAl remains atomic with maximally 1 per cent in Al-bearingmolecules which is mainly AlO. Secondly, all available Mgremains atomic and no MgO-clusters can exist. Hence,the most favoured nucleation candidates, according tothe closed models, are non-existent. Only TiO -clustersexist in the comprehensive model, with similar formationconditions as in the closed model. SiO-clusters are againdiscarded due to their low formation temperature (Sec. 4.2).The results from the comprehensive nucleation modelsuggest that TiO is the only possible AGB dust precursorof the considered nucleation candidates. However, thiscontradicts the substantial amount of Al O -favouringevidence. Firstly, the number of Al O -clusters found inpre-solar AGB grains far exceeds the amount of TiO -clusters. Secondly, numerous AGB dust observationsindicate that dust already exists close to the star and thusat temperatures as high as to , a regime inwhich, according to our model results, only Al O -clusterscan exist. TiO -clusters require temperatures below to . We believe that this discrepancy suggests that ourcurrent chemical reaction network is incomplete. Addition-ally, since there is experimental evidence that gaseous smallAl O -clusters can exists, we believe that either the currentreaction rate coefficients involving AlO-bearing moleculesare not accurate enough and need to be re-evaluated, orthat alternative small Al O -cluster formation pathwaysare missing. Moreover, most Al-molecule formation ratecoefficients are unavailable in the literature and rely on theassumption of detailed balance with their correspondingdestruction process. We therefore urge the scientific com-munity to investigate rate coefficients of formation reactionsof Al-bearing molecules at high temperatures. Without thisdata, it will remain unclear which species will form theinitial dust precursors in AGB winds.This paper has constructed and investigated an im-proved nucleation theory for more accurate modellingof the formation of dust. The improved description istime-dependent, allows growth by polymers, and considersquantum mechanical molecular properties. This procedureis universal and can be applied to any astrophysicalenvironment, where this paper focuses on AGB winds. MNRAS000
10 000 K , Boulangier et al. 2019).At around to , large MgO-clusters can form andonly at to large TiO -clusters arise. Formationof SiO-clusters is not favourable in the considered temper-ature range but requires colder conditions. Note that theabove conclusions are drawn on the underlying assumptionthat the monomer exists (Sec. 4.1).The comprehensive chemical nucleation model yieldsdifferent results from the closed nucleation ones. Firstly,it does not predict any Al O -clusters, nor its monomer,nor its molecular precursors (Al O and AlO ) but mostAl remains atomic with maximally 1 per cent in Al-bearingmolecules which is mainly AlO. Secondly, all available Mgremains atomic and no MgO-clusters can exist. Hence,the most favoured nucleation candidates, according tothe closed models, are non-existent. Only TiO -clustersexist in the comprehensive model, with similar formationconditions as in the closed model. SiO-clusters are againdiscarded due to their low formation temperature (Sec. 4.2).The results from the comprehensive nucleation modelsuggest that TiO is the only possible AGB dust precursorof the considered nucleation candidates. However, thiscontradicts the substantial amount of Al O -favouringevidence. Firstly, the number of Al O -clusters found inpre-solar AGB grains far exceeds the amount of TiO -clusters. Secondly, numerous AGB dust observationsindicate that dust already exists close to the star and thusat temperatures as high as to , a regime inwhich, according to our model results, only Al O -clusterscan exist. TiO -clusters require temperatures below to . We believe that this discrepancy suggests that ourcurrent chemical reaction network is incomplete. Addition-ally, since there is experimental evidence that gaseous smallAl O -clusters can exists, we believe that either the currentreaction rate coefficients involving AlO-bearing moleculesare not accurate enough and need to be re-evaluated, orthat alternative small Al O -cluster formation pathwaysare missing. Moreover, most Al-molecule formation ratecoefficients are unavailable in the literature and rely on theassumption of detailed balance with their correspondingdestruction process. We therefore urge the scientific com-munity to investigate rate coefficients of formation reactionsof Al-bearing molecules at high temperatures. Without thisdata, it will remain unclear which species will form theinitial dust precursors in AGB winds.This paper has constructed and investigated an im-proved nucleation theory for more accurate modellingof the formation of dust. The improved description istime-dependent, allows growth by polymers, and considersquantum mechanical molecular properties. This procedureis universal and can be applied to any astrophysicalenvironment, where this paper focuses on AGB winds. MNRAS000 , 1–68 (2019) ucleation in AGB winds This work serves as a initial model which will be extendedwith macroscopic dust formation processes such as gasaccretion, gas sputtering, dust coagulation, dust shattering,and dust evaporation in a future paper. It is the secondin a series where we strive for increased self-consistencyregarding chemistry, dust creation, and dynamics. Thedeveloped and improved chemical nucleation descriptioncan be incorporated into a hydrochemical model such as thefirst paper in this series (Boulangier et al. 2019). Currently,the results indicate which species, how much, how fast, andunder which conditions they nucleate in an AGB wind.
ACKNOWLEDGEMENTS
J.B., D.G., and L.D. acknowledge support from the ERCconsolidator grant 646758 AEROSOL. This research madeuse of Matplotlib (Hunter 2007), NumPy (Oliphant 2006),and Astropy (Robitaille et al. 2013; The Astropy Collab-oration et al. 2018), which are community-developed corePython packages for science and astronomy.
REFERENCES
Abel T., Anninos P., Zhang Y., Norman M. L., 1997, New Astron.,2, 181Archibong E. F., St-Amant A., 1999, J. Phys. Chem. A, 103, 1109Asplund M., Grevesse N., Sauval A. J., Scott P., 2009, ARA&A,47, 481Atkinson R., et al., 2004, Atmos. Chem. Phys., 4, 1461Bauschlicher C. W., Schwenke D. W., 2017, Chem. Phys. Lett.,683, 62Becke A. D., 1993, J. Chem. Phys., 98, 1372Becker K., Fink E., Leiss A., Schurath U., 1978, Chem. Phys.Lett., 54, 191Beckmann A., B¨oklen K. D., Bremer G., Elke D., 1975, Z. Phys.A Atoms Nucl., 272, 143Bojko B. T., DesJardin P. E., Washburn E. B., 2014, Combust.Flame, 161, 3211Bose M., Floss C., Stadermann F. J., Stroud R. M., SpeckA. K., 2010a, in Lunar Planet. Sci. Conf.. p. 1812, https://ui.adsabs.harvard.edu//{
Bose M., Floss C., Stadermann F. J., 2010b, ApJ, 714, 1624Bose M., Floss C., Stadermann F. J., Stroud R. M., Speck A. K.,2012, Geochim. Cosmochim. Acta, 93, 77Boulangier J., Clementel N., van Marle A. J., Decin L., de KoterA., 2019, MNRAS, 482, 5052Bromley S. T., G´omez Mart´ın J. C., Plane J. M. C., 2016, Phys.Chem. Chem. Phys., 18, 26913Burcat A., Ruscic B., 2005, Technical report, Third mil-lenium ideal gas and condensed phase thermochemicaldatabase for combustion (with update from active ther-mochemical tables)., . Argonne National Laboratory (ANL),Argonne, IL, doi:10.2172/925269,
Campbell M. L., McClean R. E., 1993, J. Phys. Chem., 97, 7942Capitelli M., Coppola C. M., Diomede P., Longo S., 2007, A&A,470, 811Catoire L., Legendre J.-F., Giraud M., 2003, J. Propuls. power,19, 196Chase M. W. J., 1998, Phys. Chem. Ref. Data, 9Chen M., Felmy A. R., Dixon D. A., 2014, J. Phys. Chem. A, 118,3136 Choi B. G., Huss G. R., Wasserburg G. J., Gallino R., 1998,Science, 282, 1284Clouet E., 2009, in Furrer D., Semiatin S., eds, , Vol. 22, ASMHandb. Fundam. Model. Met. Process. Vol. 22A. ASM In-ternational, Chapt. Modeling o, pp 203–217, https://arxiv.org/pdf/1001.4131v2.pdf
Cor`a F., 2005, Mol. Phys., 103, 2483DeMore W., et al., 1997, JPL Publ. 97-4, pp 1 – 266Decin L., et al., 2017, A&A, 608, 55Decin L., Danilovich T., Gobrecht D., Plane J. M. C., RichardsA. M. S., Gottlieb C. A., Lee K. L. K., 2018, ApJ, 855, 113Dell’Agli F., Garc´ıa-Hern´andez D. A., Schneider R., Ventura P.,La Franca F., Valiante R., Marini E., Di Criscienzo M., 2017,MNRAS, 467, 4431Demyk K., van Heijnsbergen D., von Helden G., Meijer G., 2004,A&A, 420, 547Depew K., Speck A., Dijkstra C., 2006, ApJ, 640, 971Diemand J., Ang´elil R., Tanaka K. K., Tanaka H., 2013, J. Chem.Phys., 139, 074309Farrow M. R., Chow Y., Woodley S. M., 2014, Phys. Chem. Chem.Phys., 16, 21119Ferrarotti A. S., Gail H.-P., 2006, A&A, 447, 553Forrey R. C., 2013, ApJ, 773, L25Frisch M. J., et al., 2013, Gaussian 09, Revision E.01, http://gaussian.com/
Furtenbacher T., Szidarovszky T., Hrub´y J., Kyuberis A. A.,Zobov N. F., Polyansky O. L., Tennyson J., Cs´asz´ar A. G.,2016, J. Phys. Chem. Ref. Data, 45, 043104Gail H.-P., Sedlmayr E., 1988, A&A, 206, 153Gail H.-P., Sedlmayr E., 1999, A&A, 347, 594Gamache R. R., et al., 2017, J. Quant. Spectrosc. Radiat. Transf.,203, 70Glover S. C., Abel T., 2008, MNRAS, 388, 1627Glover S. C., Federrath C., Low M. M., Klessen R. S., 2010, MN-RAS, 404, 2Gobrecht D., Cherchneff I., Sarangi A., Plane J. M. C., BromleyS. T., 2016, A&A, 585, A6Gobrecht D., Cristallo S., Piersanti L., Bromley S. T., 2017, ApJ,840, 117Gobrecht D., Decin L., Cristallo S., Bromley S. T., 2018, Chem.Phys. Lett., 711, 138Gordon I., et al., 2017, J. Quant. Spectrosc. Radiat. Transf., 203,3Goumans T. P. M., Bromley S. T., 2012, MNRAS, 420, noGrassi T., Bovino S., Schleicher D. R., Prieto J., Seifried D., Si-moncini E., Gianturco F. A., 2014, MNRAS, 439, 2386Grassi T., Bovino S., Haugbølle T., Schleicher D. R. G., 2017,MNRAS, 466, 1259Habing H., Olofsson H., 2004, Asymptotic giant branchstars, doi:10.1007/978-1-4757-3876-6. , https://ui.adsabs.harvard.edu/abs/2004agbs.book.....H
Haris K., Kramida A., 2017, ApJS, 233, 16Helling C., Winters J. M., 2001, A&A, 366, 229Helling C., Woitke P., 2006, A&A, 455, 325Herzberg G., 1966, Molecular spectra and molecular structure.Vol.3: Electronic spectra and electronic structure of poly-atomic molecules. Van Nostrand Reinhold Company, https://ui.adsabs.harvard.edu/abs/1966msms.book.....H
Higuchi Y., Fukuda Y., Fujita Y., Yamakita N., Imajo T., 2008,Chem. Phys. Lett., 452, 245H¨ofner S., Bladh S., Aringer B., Ahuja R., 2016, A&A, 594, A108Huang Y., Risha G. A., Yang V., Yetter R. A., 2009, Combust.Flame, 156, 5Huber K. P., Herzberg G., 1979, in , Mol. Spectra Mol.Struct.. Van Nostrand Reinhold Company, New York, pp8–689, doi:10.1007/978-1-4757-0961-2 2, https://ui.adsabs.harvard.edu/{%}5C{
Hunter J. D., 2007, Comput. Sci. Eng., 9, 90MNRAS , 1–68 (2019) J. Boulangier et al.
Hutcheon I. D., Huss G. R., Fahey A. J., Wasserburg G. J., 1994,ApJ, 425, L97Hynes K. M., Gyngard F., 2009, Technical report, The presolargrain database, http://presolar.wustl.edu/{~}pgd. . Labo-ratory for Space Sciences and Department of Physics, http://presolar.wustl.edu/{~}pgd.
Jacobson M. Z., 2013, Fundamentals of Atmospheric Mod-eling, 2nd edn. Cambridge University Press, New York( arXiv:1011.1669v3 ), doi:10.1017/CBO9781107415324.004Janev R., Langer W., Evans K., 1987, Elementary processes inHydrogen-Helium plasmas - Cross sections and reaction ratecoefficients. Springer, BerlinJeong K. S., Chang C., Sedlmayr E., S¨ulzle D., 2000, J. Phys. BAt. Mol. Opt. Phys., 33, 3417Jeong K. S., Winters J. M., Le Bertre T., Sedlmayr E., 2003,A&A, 407, 191Johns J. W. C., Priddle S. H., Ramsay D. A., 1963, Discuss.Faraday Soc., 35, 90Johnson R. D. I., 2018, NIST Computational Chemistry Com-parison and Benchmark Database NIST Standard Refer-ence Database Number 101, doi:10.18434/T47C7Z, http://cccbdb.nist.gov/
Johnston R. L., 2002, Atomic and Molecular Clus-ters, 1st edn. Master’s Series in Physics and As-tronomy, CRC Press, doi:10.1201/9781420055771,
Kami´nski T., et al., 2016, A&A, 592, A42Kami´nski T., et al., 2017, A&A, 599, A59Karakas A. I., 2010, MNRAS, 403, 1413Karakas A., Lattanzio J. C., 2007, Publ. Astron. Soc. Aust., 24,103Khouri T., et al., 2016, A&A, 591, A70Khouri T., Vlemmings W. H. T., Olofsson H., Ginski C., De BeckE., Maercker M., Ramstedt S., 2018, A&A, 620, 75K¨ohler T. M., Gail H.-P., Sedlmayr E., 1997, A&A, 320, 553Kramida A., Ralchenko Y., Reader J., NIST ASD Team (2018)2018, Natl. Inst. Stand. Technol.Kurucz R., 1992, Rev. Mex. Astron. y Astrofis., 23Lamiel-Garcia O., Cuko A., Calatayud M., Illas F., Bromley S. T.,2017, Nanoscale, 9, 1049Langowski M. P., et al., 2015, Atmos. Chem. Phys, 15, 273Lee G., Helling C., Giles H., Bromley S. T., 2015, A&A, 575, A11Lee G. K. H., Blecic J., Helling C., 2018, A&A, 614, 126Lepinoux J., 2006, Philos. Mag., 86, 5053Li R., Cheng L., 2012, Comput. Theor. Chem., 996, 125Li G., Gordon I. E., Rothman L. S., Tan Y., Hu S.-M., Kassi S.,Campargue A., Medvedev E. S., 2015, ApJS, 216, 15Luo Y., 2007, Comprehensive Handbook of Chemical Bond En-ergies. CRC Press, http://staff.ustc.edu.cn/{~}luo971/2010-91-CRC-BDEs-Tables.pdf
Martin W. C., Zalubas R., 1979, J. Phys. Chem. Ref. Data, 8,817Martin W. C., Zalubas R., 1983, J. Phys. Chem. Ref. Data, 12,323Mart´ınez-N´u˜nez E., Whalley C. L., Shalashilin D., Plane J. M. C.,2010, J. Phys. Chem. A, 114, 6472McElroy D., Walsh C., Markwick A. J., Cordiner M. A., SmithK., Millar T. J., 2013, A&A, 550, A36McKinnon R., Vogelsberger M., Torrey P., Marinacci F., KannanR., 2018, MNRAS, 478, 2851McSween H. J., Huss G. R., 2010, Cosmochemistry. CambridgeUniversity Press, doi:10.1111/j.1945-5100.2011.01192.x, https://books.google.be/books?id=385nPZOXmYAC
Moore C. E., 1993, in Gallagher J. W., ed., , CRC Ser. Eval. DataAt. Phys.. CRC Press, Boca Raton, FLNguyen A. N., Messenger S., 2009, Identification of anExtremely 18O-rich Presolar Silicate Grain in Acfer094, https://ui.adsabs.harvard.edu/{
Nittler L. R., Alexander C. M. O., Gao X., Walker R. M., ZinnerE. K., 1994, Nature, 370, 443Nittler L. R., Alexander C. M. O., Gao X., Walker R. M., ZinnerE., 1997, ApJ, 483, 475Nittler L. R., Alexander C. M. O., Gallino R., Hoppe P., NguyenA. N., Stadermann F. J., Zinner E. K., 2008, ApJ, 682, 1450Norris B. R. M., et al., 2012, Nature, 484, 220Ohnaka K., Weigelt G., Hofmann K.-H., 2016, A&A, 589, A91Oliphant T. E., 2006, A guide to NumPyPatrascu A. T., Yurchenko S. N., Tennyson J., 2015, MNRAS,449, 3613Patzer A. B. C., Gauger A., Sedlmayr E., 1998, A&A, 337, 847Patzer A. B., Chang C., Sedlmayr E., S¨ulzle D., 2005, Eur. Phys.J. D, 32, 329Phillips J. G., 1971, ApJ, 169, 185Plane J. M. C., 2013, Philos. Trans. A. Math. Phys. Eng. Sci.,371, 20120335Plane J. M. C., Helmer M., 1995, Faraday Discuss., 100, 411Plane J. M. C., Whalley C. L., 2012, J. Phys. Chem. A, 116, 6240Plane J. M. C., Feng W., Dawkins E. C. M., 2015, Chem. Rev.,115, 4497Popovas A., Jørgensen U. G., 2016, A&A, 595, A130Posch T., Kerschbaum F., Mutschke H., Fabian D., Dorschner J.,Hron J., 1999, A&A, 352, 609Poulaert G., Brouillard F., Claeys W., McGowan J. W., Wassen-hove G. V., 1978, J. Phys. B At. Mol. Phys., 11, L671Ritter D., Weisshaar J. C., 1989, J. Phys. Chem, 93, 1576Robitaille T. P., et al., 2013, A&A, 558, A33Rollason R. J., Plane J. M. C., 2001, Phys. Chem. Chem. Phys.,3, 4733Rothman L., et al., 2010, J. Quant. Spectrosc. Radiat. Transf.,111, 2139Saloman E. B., 2012, J. Phys. Chem. Ref. Data, 41, 013101Sarangi A., Cherchneff I., 2015, A&A, 575, A95Savel’ev A. M., Starik A. M., 2018, Combust. Flame, 196, 223Sharipov A. S., Loukhovitski B. I., 2018, Combust. Explos., 11Sharipov A. S., Starik A. M., 2016, Chem. Phys., 465-466, 9Sharipov A., Titova N., Starik A., 2011, J. Phys. Chem. A, 115,4476Sharipov A. S., Titova N. S., Starik A. M., 2012, Combust. TheoryModel., 16, 842Sierka M., et al., 2007, Angew. Chemie Int. Ed., 46, 3372Sloan G. C., Levan P. D., Little-Marenin I. R., 1996, ApJ, 463,310Sloan G. C., Kraemer K. E., Goebel J. H., Price S. D., 2003, ApJ,594, 483Sluder A., Milosavljevi´c M., Montgomery M. H., 2018, MNRAS,480, 5580Speck A. K., Barlow M. J., Sylvester R. J., Hofmeister A. M.,2000, A&AS, 146, 437Starik A. M., Kuleshov P. S., Sharipov A. S., Titova N. S., TsaiC.-J., 2014, Combust. Flame, 161, 1659Starik A. M., Savel’ev A. M., Titova N. S., 2015, Combust. Explos.Shock Waves, 51, 197Stroud R. M., Nittler L. R., Alexander C. M. O., 2004, Science,305, 1455Stroud R. M., Nittler L. R., Alexander C. M. O., Zinner E., 2007,Lunar Planet. Sci. Conf., p. 2203Suh S.-M., Zachariah M. R., Girshick S. L., 2001, J. Vac. Sci.Technol. A Vacuum, Surfaces, Film., 19, 940Swihart M. T., Catoire L., Legrand B., G¨okalp I., Paillard C.,2003, Combust. Flame, 132, 91Takigawa A., Tachibana S., Nagahara H., Ozawa K., 2015, ApJS,218Tanaka K. K., Tanaka H., Yamamoto T., Kawamura K., 2011, J.Chem. Phys., 134, 204313 MNRAS000
Nittler L. R., Alexander C. M. O., Gao X., Walker R. M., ZinnerE. K., 1994, Nature, 370, 443Nittler L. R., Alexander C. M. O., Gao X., Walker R. M., ZinnerE., 1997, ApJ, 483, 475Nittler L. R., Alexander C. M. O., Gallino R., Hoppe P., NguyenA. N., Stadermann F. J., Zinner E. K., 2008, ApJ, 682, 1450Norris B. R. M., et al., 2012, Nature, 484, 220Ohnaka K., Weigelt G., Hofmann K.-H., 2016, A&A, 589, A91Oliphant T. E., 2006, A guide to NumPyPatrascu A. T., Yurchenko S. N., Tennyson J., 2015, MNRAS,449, 3613Patzer A. B. C., Gauger A., Sedlmayr E., 1998, A&A, 337, 847Patzer A. B., Chang C., Sedlmayr E., S¨ulzle D., 2005, Eur. Phys.J. D, 32, 329Phillips J. G., 1971, ApJ, 169, 185Plane J. M. C., 2013, Philos. Trans. A. Math. Phys. Eng. Sci.,371, 20120335Plane J. M. C., Helmer M., 1995, Faraday Discuss., 100, 411Plane J. M. C., Whalley C. L., 2012, J. Phys. Chem. A, 116, 6240Plane J. M. C., Feng W., Dawkins E. C. M., 2015, Chem. Rev.,115, 4497Popovas A., Jørgensen U. G., 2016, A&A, 595, A130Posch T., Kerschbaum F., Mutschke H., Fabian D., Dorschner J.,Hron J., 1999, A&A, 352, 609Poulaert G., Brouillard F., Claeys W., McGowan J. W., Wassen-hove G. V., 1978, J. Phys. B At. Mol. Phys., 11, L671Ritter D., Weisshaar J. C., 1989, J. Phys. Chem, 93, 1576Robitaille T. P., et al., 2013, A&A, 558, A33Rollason R. J., Plane J. M. C., 2001, Phys. Chem. Chem. Phys.,3, 4733Rothman L., et al., 2010, J. Quant. Spectrosc. Radiat. Transf.,111, 2139Saloman E. B., 2012, J. Phys. Chem. Ref. Data, 41, 013101Sarangi A., Cherchneff I., 2015, A&A, 575, A95Savel’ev A. M., Starik A. M., 2018, Combust. Flame, 196, 223Sharipov A. S., Loukhovitski B. I., 2018, Combust. Explos., 11Sharipov A. S., Starik A. M., 2016, Chem. Phys., 465-466, 9Sharipov A., Titova N., Starik A., 2011, J. Phys. Chem. A, 115,4476Sharipov A. S., Titova N. S., Starik A. M., 2012, Combust. TheoryModel., 16, 842Sierka M., et al., 2007, Angew. Chemie Int. Ed., 46, 3372Sloan G. C., Levan P. D., Little-Marenin I. R., 1996, ApJ, 463,310Sloan G. C., Kraemer K. E., Goebel J. H., Price S. D., 2003, ApJ,594, 483Sluder A., Milosavljevi´c M., Montgomery M. H., 2018, MNRAS,480, 5580Speck A. K., Barlow M. J., Sylvester R. J., Hofmeister A. M.,2000, A&AS, 146, 437Starik A. M., Kuleshov P. S., Sharipov A. S., Titova N. S., TsaiC.-J., 2014, Combust. Flame, 161, 1659Starik A. M., Savel’ev A. M., Titova N. S., 2015, Combust. Explos.Shock Waves, 51, 197Stroud R. M., Nittler L. R., Alexander C. M. O., 2004, Science,305, 1455Stroud R. M., Nittler L. R., Alexander C. M. O., Zinner E., 2007,Lunar Planet. Sci. Conf., p. 2203Suh S.-M., Zachariah M. R., Girshick S. L., 2001, J. Vac. Sci.Technol. A Vacuum, Surfaces, Film., 19, 940Swihart M. T., Catoire L., Legrand B., G¨okalp I., Paillard C.,2003, Combust. Flame, 132, 91Takigawa A., Tachibana S., Nagahara H., Ozawa K., 2015, ApJS,218Tanaka K. K., Tanaka H., Yamamoto T., Kawamura K., 2011, J.Chem. Phys., 134, 204313 MNRAS000 , 1–68 (2019) ucleation in AGB winds Teyssandier F., Allendorf M. D., 1998, J. Electrochem. Soc., 145,2167The Astropy Collaboration T. A., et al., 2018, ArXiv, 156, 123Toxvaerd S., 2015, J. Chem. Phys., 143, 154705Tsai S.-M., Lyons J. R., Grosheintz L., Rimmer P. B., KitzmannD., Heng K., 2017, ApJS, 228, 20Verner D. A., Ferland G. J., 1996, ApJS, 103, 467Vidler M., Tennyson J., 2000, J. Chem. Phys., 113, 9766Wakelam V., et al., 2012, ApJS, 199, 21Washburn E. B., Trivedi J. N., Catoire L., Beckstead M. W., 2008,Combust. Sci. Technol., 180, 1502Whalley C. L., Plane J. M. C., 2010, Faraday Discuss., 147, 349Whalley C. L., Mart´ın J. C. G., Wright T. G., Plane J. M. C.,2011, Phys. Chem. Chem. Phys., 13, 6352Woitke P., 2006a, A&A, 452, 537Woitke P., 2006b, A&A, 460, 9Wong A., Yurchenko S. N., Bernath P., M¨uller H. S. P., McConkeyS., Tennyson J., 2017, MNRAS, 470, 882Yurchenko S. N., Williams H., Leyland P. C., Lodi L., TennysonJ., 2018, MNRAS, 479, 1401Zeidler S., Posch T., Mutschke H., 2013, A&A, 553, 81Zhao-Geisler R., K¨ohler R., Kemper F., Kerschbaum F., MayerA., Quirrenbach A., Lopez B., 2015, Publ. Astron. Soc. Pa-cific, 127, 732van Heijnsbergen D., Demyk K., Duncan M. A., Meijer G., vonHelden G., 2003, Phys. Chem. Chem. Phys., 5, 2515
APPENDIX A: EQUILIBRIUM COMPOSITIONIN THE DILUTE LIMIT
This section describes, step by step, how to determine theequilibrium composition of a gas mixture. This allows to de-termine the equilibrium ratio of two species, which in neededin equation (13). We focus on a nucleating system as this isthe main purpose of this work. Because the number densi-ties of nucleating molecules are small compared to the totalgas number density, a nucleating system can be consideredas a dilute solution where the bulk gas is the solvent andthe nucleation molecules are the solutes. The Gibbs free en-ergy of a pure solvent of N A particles A , is just N A times thechemical potential, G = N A µ A ( T , P ) , (A1)where µ A ( T , P ) is the chemical potential of the pure solvent,that is a function of temperature and pressure, T and P .Imagine, adding a single B particle to this system whileholding the temperature and pressure fixed. This changesthe Gibbs free energy by d G = d U + P d V − T d S , (A2)where U is the internal energy, V the volume, and S the en-tropy of the system. Note that d U nor P d V depend on N A but on how the B particle interacts with its nearby neigh-bours, regardless of the total number of A particles. d S ispartly independent of N A , but part comes from the freedomof choosing where to put this B particle. As this is propor-tional to the total number of A particles, the entropy changesas d S = k ln N A + ( terms independent of N A ) . (A3)We drop the B subscript of the Boltzmann constant to avoidconfusion with the B particle. The total change in Gibbs free energy can then be written as d G = G B ( T , P ) − kT ln N A , (A4)where G B ( T , P ) is a function of temperature and pressurebut independent of N A . We shall call this the intrinsic Gibbsfree energy of particle B . Generalising to adding N B particlesresults in a change d G = N B G B ( T , P ) − N B kT ln N A + kT ln ( N B ! ) (A5)where the last term is introduced because all B particles areidentical and interchanging them does not result in a distinctstate. Because N B (cid:29) , Stirling’s approximation can be usedto get rid of the factorial, leading to d G = N B G B ( T , P ) − N B kT ln N A + N B kT ln N B − N B kT . (A6)Generalising this to adding M − different particles (so that M includes the solvent particle), the total Gibbs free energyof the system is given by G = N A µ A ( T , P ) + M (cid:213) i = N i G i ( T , P )− N i kT ln N A + N i kT ln N i − N i kT . (A7)Note that this expression is only valid in the limit N i (cid:28) N A ,that is when the solution is dilute. If not, then all i particleswould also interact with each other and the volume occupiedby the particles will matter in the total Gibbs free energydetermination (Lepinoux 2006). In order to determine theequilibrium composition of the system, its Gibbs free energy(Eq. A7) has to be minimised.In general, when optimising a multivariate function f ( x , . . . , x n ) with m number of constraints g k ( x , . . . , x n ) = with k ∈ { , . . . , m } , the Lagrangian that needs to beoptimised (to each variable) takes the form L( x , . . . , x n , λ , . . . , λ m ) = f ( x , . . . , x n ) − m (cid:213) k = λ k g k ( x , . . . , x n ) , (A8)where each λ k is called a Lagrangian multiplier. Minimisingthe total Gibbs free energy of the nucleating system, eq.(A7), can be achieved under the constraint that the totalnumber of atoms in the system is constant M (cid:213) i = A (cid:213) j = N i x ij = C , (A9)where C is the total number of atoms, A is the number ofdifferent atoms, and x ij is the number of j atoms in molecule i . Since this one constraint is sufficient, g k = g = g and λ k = λ = λ . Rewriting the constraint gives g ( N , . . . , N M ) = M (cid:213) i = A (cid:213) j = N i x ij − C = , (A10)with N = N A (the solvent). The Lagrangian of the systemcan then be written as L( N , . . . , N M ) = G ( N , . . . , N M ) − λ (cid:169)(cid:173)(cid:171) M (cid:213) i = A (cid:213) j = N i x ij − C (cid:170)(cid:174)(cid:172) . MNRAS , 1–68 (2019) J. Boulangier et al. (A11)Minimising this Lagrangian to each variable leads to the setof M + equations ∂ L ∂ N A = µ A ( T , P ) − kTN A M (cid:213) i = N i − λ A (cid:213) j = x Aj = (A12) ∂ L ∂ N i = G i ( T , P ) + kT ln (cid:18) N i N A (cid:19) − λ A (cid:213) j = x ij = (A13) ∂ L ∂λ = M (cid:213) i = A (cid:213) j = N i x ij − C = (A14)where eq. (A13) is valid for all i ∈ { , . . . , M} . Solving thismatrix will result in the equilibrium distribution of allmolecules.For our purpose (making use of detailed balance, eq.13), we are interested in the ratio between molecules andby rewriting equation (A13) the number of solute moleculescompared to the solvent is given by N i N A = exp (cid:169)(cid:173)(cid:171) − G i + λ (cid:205) A j x ij kT (cid:170)(cid:174)(cid:172) . (A15)Note this represents the equilibrium values but we omit the”eq” superscript for clarity. As we consider a nucleating sys-tem, this equation can be simplified, because the number ofatoms in a cluster scales linearly with the size of the cluster.Let X be the number of atoms in the monomer, then for acluster of size n : A (cid:213) j x nj = n A (cid:213) j x j = nX . (A16)Then, according to equation (A15), the number fraction ofan n -sized cluster is given by N n N A = exp (cid:18) − G n + λ nXkT (cid:19) . (A17)Consequently, the ratio of two different cluster sizes n and m is N n N m = exp (cid:18) − G n + G m + ( n − m ) λ XkT (cid:19) . (A18)Introducing an ( n − m ) -sized cluster, with n > m , removes the λ X term. I.e. using Eq. (A17), one can write N n − m N A = exp (cid:18) − G n − m + ( n − m ) λ XkT (cid:19) , (A19)and hence ( n − m ) λ X = kT ln (cid:18) N n − m N A (cid:19) + G n − m . (A20)Substitution Eq. (A20) into equation (A17), results in theratio, N n N m = N n − m N A exp (cid:18) − G n + G m + G n − m kT (cid:19) . (A21)Remember that each G i = G i ( T , P ) is temperature andpressure dependent. For convenience these values are often calculated at a so-called standard pressure of P ◦ = ( = · Pa = · dyne / cm ) . The superscript ◦ refers toa quantity at this standard pressure. The Gibbs free energyof a particle at any pressure can be written as a function ofthe standard one, G = G ◦ − kT ln (cid:18) P ◦ P (cid:19) , (A22)because only the translational partition function is a pres-sure dependent term (Eqs. B10–B9), Z t = Z ◦ t P ◦ P . (A23)Using the standard Gibbs free energy and substituting equa-tion (A22) in equation (A21), the ratio of cluster sizes be-comes, N n N m = N n − m N A exp (cid:18) − G ◦ n + G ◦ m + G ◦ n − m kT (cid:19) PP ◦ = N n − m kTP ◦ V exp (cid:18) − G ◦ n + G ◦ m + G ◦ n − m kT (cid:19) (A24)Hence, in equilibrium, the ratio of number densities of twoclusters of sizes N and M , with N > M , is described by, n eq N n eq M = n eq N − M kTP ◦ exp (cid:18) − G ◦ N + G ◦ M + G ◦ N − M kT (cid:19) . (A25) APPENDIX B: GIBBS FREE ENERGY
The Gibbs free energy of a system is defined as G = H − T S , (B1)where H is the enthalpy, S is the entropy, and T is the tem-perature of the system. The enthalpy is defined as H = U + PV , (B2)where U is the internal energy of the system, P is the pres-sure of the system, and V is the volume of the system. Bothentropy and internal energy depend on the configurationalfreedom of the particles in the system. This configurationalfreedom or statistical properties of a particle is describedby its partition function. When dealing with a system of N non-interacting particles, the system’s partition function isgiven by Z N = N ! Z N , (B3)where Z is the partition function of a single particle.The entropy for a system consisting of N particles isdefined as S N = ∂ kT ln Z N ∂ T (cid:12)(cid:12)(cid:12)(cid:12) V , N = k ln Z N + kT ∂ ln Z N ∂ T (cid:12)(cid:12)(cid:12)(cid:12) V , N (B4) MNRAS000
The Gibbs free energy of a system is defined as G = H − T S , (B1)where H is the enthalpy, S is the entropy, and T is the tem-perature of the system. The enthalpy is defined as H = U + PV , (B2)where U is the internal energy of the system, P is the pres-sure of the system, and V is the volume of the system. Bothentropy and internal energy depend on the configurationalfreedom of the particles in the system. This configurationalfreedom or statistical properties of a particle is describedby its partition function. When dealing with a system of N non-interacting particles, the system’s partition function isgiven by Z N = N ! Z N , (B3)where Z is the partition function of a single particle.The entropy for a system consisting of N particles isdefined as S N = ∂ kT ln Z N ∂ T (cid:12)(cid:12)(cid:12)(cid:12) V , N = k ln Z N + kT ∂ ln Z N ∂ T (cid:12)(cid:12)(cid:12)(cid:12) V , N (B4) MNRAS000 , 1–68 (2019) ucleation in AGB winds Substituting Z N using equation (B3) yields, S N = N k ln Z − k ln ( N ! ) + kT ∂ N ln Z − k ln ( N ! ) ∂ T (cid:12)(cid:12)(cid:12)(cid:12) V , N = N k ln Z + N kT ∂ ln Z ∂ T (cid:12)(cid:12)(cid:12)(cid:12) V − k ln ( N ! ) (B5) = NS − k ln ( N ! )≈ NS − N k ln N + kN , where the last transition uses Stirling’s approximationwhich is valid for N (cid:29) . As this quantity is often calculatedfor one mole ( .
022 140 758 · particles), this is a validapproximation.The internal energy of a system consisting of N parti-cles is defined as U N = kT ∂ ln Z N ∂ T (cid:12)(cid:12)(cid:12)(cid:12) V , N (B6)Again, substituting Z N with equation (B3), this reduces to U N = N kT ∂ ln Z ∂ T (cid:12)(cid:12)(cid:12)(cid:12) V = NU . (B7)Typically, the partition function is calculated with respectto the bottom of the particle’s energy well (Sec. B1.4) There-fore this energy value, U , is separated from the partitionfunction and equation (B7) becomes, U N = N kT ∂ ln Z ∂ T (cid:12)(cid:12)(cid:12)(cid:12) V + NU (B8) = N ( U + U ) . Substituting equations (B2), (B5), and (B8) into (B1), com-bined with the ideal gas law, yields the Gibbs free energy ofa system of N particles, G N = NU − N kT ln Z + N kT ln N , (B9)which only depends on the total partition function of a singleparticle and U of that particle. B1 Partition functions of one particle
According to the Born-Oppenheimer approximation rota-tional, vibrational, and electronic energies are independentof each other, and the partition function of one particle canbe written as the product of separate contributors namelytranslational, rotational, vibrational, and electronic degreesof freedom, Z = Z tr Z rot Z vib Z el . This section contains asummary of all different partition function for the most gen-eral case of a non-linear poly atomic ideal gas, a linear polyatomic ideal gas, and a mono atomic ideal gas. U is the sum of the electronic ground state and nuclear-nuclearrepulsion energies, isolated in vacuum, without vibration at . B1.1 Translation
The translational part is always given by Z tr = (cid:18) π mkTh (cid:19) / V = (cid:18) π mkTh (cid:19) / N kTP , (B10)where m is the mass of the particle and h is the Planckconstant Note that V is the volume of the embedding systemmeaning that N is the total number of particles of the systemin which this one particle resides. B1.2 Rotation (I) Non-linear poly atomic Z rot = σ (cid:18) π T Θ x Θ y Θ z (cid:19) / , (B11)where σ is the molecule’s symmetry number , and Θ i the rotational temperature related to the moments ofinertia, I x , I y , I z , via Θ i = (cid:126) I i k i ∈ { x , y , z } (B12)(II) Linear poly atomic Z rot = T Θ rot σ , (B13)where Θ rot is the rotational temperature related tothe moment of inertia, I via Θ = (cid:126) I k (B14)(III) Mono atomic Z rot = (B15)Note that this is a high temperature approximation which isvalid when the temperature is much larger than rotationaltemperature, which is the case in all our simulations. B1.3 Vibration
A molecules consisting of N atoms has N degrees of free-dom, where the factor ”3” corresponds to the possible move-ments of a particle in three-dimensional space. In the mostgeneral case, a molecule has N − − = N − vibrationaldegrees of freedom where the ” − ” terms are the transla-tional and rotational degrees of freedom of the molecule.We choose the zero-energy reference point as the bottom ofthe potential well and not the vibrational ground state.(1) Non-linear poly atomic Z vib = (cid:214) Θ v ∈T v e − Θ v / T − e − Θ v / T , (B16) A molecule’s symmetry number is the number of different butindistinguishable views of the molecule to correct for countingequivalent views.MNRAS , 1–68 (2019) J. Boulangier et al. where Θ v is the vibrational temperature related to avibrational frequency ν of the molecule via Θ v = h ν k (B17)when assuming that the vibrational modes of themolecule behave like harmonic oscillators. T v is the setof all N − vibrational modes of the molecule.(2) Linear poly atomic Z vib = (cid:214) Θ v ∈T v e − Θ v / T − e − Θ v / T . (B18)Note that T v only contains N − vibrational modesdue to a rotational symmetry of the molecule.(3) Mono atomic Z vib = (B19) B1.4 Electronic
The electronic part is always given by Z el = N e (cid:213) i = g i e − (cid:15) i / kT (B20)with (cid:15) i the i th electronic energy level w.r.t. the bottom of theelectronic potential well, g i the degeneracy of the i th leveldue to spin splitting and N e the number of energy levels.Each energy level can be scaled by choosing the bottom ofthe well to be 0 , giving ε i = (cid:15) i − (cid:15) . The number of levelscan also be limited to the one where ε N lim (cid:29) kT . Z el = g + N lim (cid:213) i = g i e − ε i / kT (B21) APPENDIX C: GIBBS FREE ENERGY OFFORMATION
Generally, the standard Gibbs free energy of formation(GFEoF), rather than the Gibbs free energy (GFE), isused to determine reversed reaction rate coefficients underthe assumption of detailed balance. Although both can beused, we opt for GFE for reason explained in the maintext (Sec. 3.6.2) but explain GFEoF for completeness andcomparison. The GFEoF of a compound is the change inGFE that occurs when one mole of the compound is formedfrom its component elements in their most thermodynam-ically stable states under standard conditions (pressure of1 bar = · Pa ). Note that this state, depending on thecomponents can be gaseous, solid, or liquid.Consider a molecule m consisting of N unique atomswith each atom a occurring v a times in the molecule. Then,the set with unique atoms is defined as A = { a , a , . . . , a N } .For an example molecule m = H O, this gives N = , This energy difference should be added again in the total in-ternal energy of the molecule (Eq. B8.) A = { H , O } , v H = , and v O = . Following the documen-tation of gaussian09 (Frisch et al. 2013), the standardGFEoF of molecule m at a given temperature T , ∆ f G ◦ T , m , isdescribed by ∆ f G ◦ T , m = ∆ f H ◦ T , m − T (cid:32) S ◦ T , m − (cid:213) a ∈A v a S ◦ T , a (cid:33) , (C1)where ∆ f H ◦ T , m is the standard enthalpy of formation ofmolecule m at a given temperature, S ◦ T , m and S ◦ T , a are theentropy at a given temperature of molecule m and atom a ,respectively. The ◦ notation refers to the quantity at stan-dard pressure of 1 bar (= · Pa ). The standard enthalpyof formation of molecule m at temperature T is described by ∆ f H ◦ T , m = ∆ f H ◦ , m + H ◦ T , m − H ◦ , m − (cid:213) a ∈A v a (cid:16) H ◦ T , a − H ◦ , a (cid:17) , (C2)where H ◦ T denotes the standard (thermal) enthalpy (Eq. B2)which excludes the electronic potential energy U of thespecies. The standard enthalpy of formation of a moleculeat absolute zero is given by ∆ f H ◦ , m = U , m + U zpve , m − (cid:213) a ∈A v a (cid:16) U , a − ∆ f H ◦ , a (cid:17) , (C3)where U zpve , m is the zero point vibration energy of amolecule, which is the lowest vibrational energy (groundstate) at . Note that this is not the bottom of the vi-brational potential well (when representing this as harmonicoscillator potential). Combining equations C1, C2, and C3,and rearranging some terms, the standard GFEoF is givenby ∆ f G ◦ T , m = H ◦ T , m − H ◦ , m + U , m + U zpve , m − T S ◦ T , m − (cid:213) a ∈A v a (cid:16) H ◦ T , a − H ◦ , a + U , a − ∆ f H ◦ , a − T S ◦ T , a (cid:17) . (C4)When realising that H ◦ , m = U zpve , m for a molecule and H ◦ , a = for an atom, the standard GFEoF reduces to ∆ f G ◦ T , m = H ◦ T , m + U , m − T S ◦ T , m − (cid:213) a ∈A v a (cid:16) H ◦ T , a + U , a − ∆ f H ◦ , a − T S ◦ T , a (cid:17) . (C5) APPENDIX D: QUANTUM MECHANICALDATA
This section contains an overview of all quantum mechan-ical data that was collected and calculated (Table D1 forthe nucleation species and Table D2 for all other species).All gathered data has been homogenised and is available asa JSON file. A collection of used literature input files (raw The standard enthalpy of formation of a compound is thechange of enthalpy during the formation of one mole of that sub-stance from its constituent elements, with all substances in theirstandard states. For an atom, this is the standard enthalpy ofphase transition w.r.t. the phase in its standard state, i.e. the en-ergy that must be supplied as heat at constant pressure per moleto convert from one phase to the other.MNRAS , 1–68 (2019) ucleation in AGB winds Table D1.
Nucleation cluster specifications. All quantum me-chanical properties of these clusters are calculated in this work( U , Z / Z tr , Θ rot , Θ vib ).Cluster Sizes Global minimum r monomer (nm)TiO a SiO 1-10 Bromley et al. (2016) 0.075765 b MgO 1-10 Chen et al. (2014) 0.0865 c Al O d Notes: a) Inter atomic Ti O distance from Jeong et al. (2000).b) Half a Si O bond length from Bromley et al. (2016). c) Half aMg O bond length from Farrow et al. (2014). d) Inter atomic dis-tance O Al O (linear geometry) from Archibong & St-Amant(1999). All used monomer radii can be more accurate by account-ing for the geometry of the non-linear molecules and using ourre-evaluated structures (Sec. 6.1.1). and cleaned versions), reference files, and info files is alsoavailable online . All this data was used to calculate Gibbsfree energies, which are also available online for the tem-perature range on our interest to at standardpressure of , which also have been included in KROME.Adaptations of these tables can easily be produced with ouropen-source repository and the provided data. Zenodo: https://zenodo.org/record/3356710 https://bitbucket.org/JelsB/thermochemistry APPENDIX E: RESULTS
This appendix encompasses additional figures of the nucle-ation models. Figures which are not shown in this appendixare either already present in the main body or provide noadded value.
E1 Closed nucleation networks
This section contains a more thorough overview of all closednucleation models of all nucleation clusters results.
E1.1 Monomer nucleation
This section contains a more complete overview of theclosed nucleation models using the monomer nucleationdescription of all nucleation clusters results.TiO -clusters: Figs. E1 to E2MgO-clusters: Figs. E3 to E5SiO-clusters: Fig. E6Al O -clusters: Figs. E7 to E9 E1.2 Polymer nucleation
This section contains a more complete overview of theclosed nucleation models using the polymer nucleationdescription of all nucleation clusters results.TiO -clusters: Figs. E10 to E11MgO-clusters: Figs. E12 to E14SiO-clusters: Fig. E15Al O -clusters: Figs. E16 to E18 E1.3 Polymer nucleation compared with equilibrium
This section contains figures which compare the relative ra-tios of nucleation clusters of the closed nucleation modelsw.r.t. the equilibrium ratios (Figs. E19 to E21).
E2 Comprehensive chemical nucleation networks
This section contains a more complete overview of all nu-cleation clusters results in the comprehensive chemical nu-cleation model using the polymer nucleation description.No Mg-related figures are shown as it remains completelyatomic.
E2.1 Ti-bearing species
This section contains a more complete overview of all Ti-bearing species results in the comprehensive chemical nu-cleation model using the polymer nucleation description(Figs. E22, E23).
E2.2 Si-bearing species
This section contains a more complete overview of all Si-bearing species results in the comprehensive chemical nu-cleation model using the polymer nucleation description(Fig. E24).
MNRAS , 1–68 (2019) J. Boulangier et al.
Table D2.
Overview of the sources of all quantum mechanical data, either gathered or calculated, as defined in Appendix B.Species Global minimum U Z / Z tr Θ rot , Θ vib ε i TiO - CCCBDB Kurucz (1992) a - Phillips (1971) c CO - CCCBDB Rothman et al. (2010) a - Herzberg (1966) e OH - CCCBDB Rothman et al. (2010) a - Huber & Herzberg (1979) e AlO - CCCBDB Patrascu et al. (2015) a - -AlH - CCCBDB Yurchenko et al. (2018) a - -NO - CCCBDB Wong et al. (2017) a - Huber & Herzberg (1979) e CO - CCCBDB Li et al. (2015) a - -SO - CCCBDB Gamache et al. (2017) b - ? c SO - CCCBDB Gamache et al. (2017) b - Herzberg (1966) e HO - CCCBDB Gamache et al. (2017) b - -H O - CCCBDB Gamache et al. (2017) b - -O - CCCBDB Gamache et al. (2017) b - Huber & Herzberg (1979) e N - CCCBDB Gamache et al. (2017) b - -N O - CCCBDB Gamache et al. (2017) b - Herzberg (1966) e NO - CCCBDB Gamache et al. (2017) b - ? e H O - CCCBDB Furtenbacher et al. (2016) f - -H - CCCBDB Popovas & Jørgensen (2016) - Huber & Herzberg (1979) e AlC - CCCBDB CCCBDB -AlH - CCCBDB CCCBDB -AlH - CCCBDB CCCBDB -HCO - CCCBDB CCCBDB Johns et al. (1963) e , g HO - CCCBDB CCCBDB Becker et al. (1978) e , h MgO - CCCBDB CCCBDB Bauschlicher & Schwenke (2017) i , Huber & Herzberg (1979) e MgOH - CCCBDB CCCBDB -Mg(OH) - CCCBDB CCCBDB -MgCO - CCCBDB CCCBDB -O - CCCBDB CCCBDB -SiO - CCCBDB CCCBDB -AlO Patzer et al. (2005) -Al O Patzer et al. (2005) -Al O Patzer et al. (2005) -AlOH -AlO H -Al(OH) -Al(OH) -H (cid:55) CCCBDB (cid:55) (cid:55) -C (cid:55) CCCBDB (cid:55) (cid:55)
Haris & Kramida (2017); Beckmann et al. (1975) d Mg (cid:55) (cid:55) (cid:55) -N (cid:55) (cid:55) (cid:55) -O (cid:55) (cid:55) (cid:55) Moore (1993) d Si (cid:55) (cid:55) (cid:55) Martin & Zalubas (1983) d Al (cid:55) (cid:55) (cid:55) Martin & Zalubas (1979) d Ti (cid:55) (cid:55) (cid:55) Saloman (2012) d Legend : : This work, (cid:55) : Not applicable, -: Unnecessary, ? : No references provided, CCCBDB: NIST Computational ChemistryComparison and Benchmark Database (Johnson 2018).a) via ExoMol ( http://exomol.com/ ). b) via HITRAN (Gordon et al. 2017). c) via NIST chemistry WebBook ( https://doi.org/10.18434/T4D303 ). d) via NIST Atomic Spectra Database (Kramida et al. 2018). e) via CCCBDB (Johnson 2018) f) Uses g = and g = as para-ortho degeneracy which is preferred over using g = / and g = / like Vidler & Tennyson (2000). g) The mostlikely reference of list of the references provided by NIST chemistry WebBook ( https://doi.org/10.18434/T4D303 ). h) Unclearreference for the second energy level. i) First level: improved theoretical value over the theoretical one of Huber & Herzberg (1979).MNRAS000
Haris & Kramida (2017); Beckmann et al. (1975) d Mg (cid:55) (cid:55) (cid:55) -N (cid:55) (cid:55) (cid:55) -O (cid:55) (cid:55) (cid:55) Moore (1993) d Si (cid:55) (cid:55) (cid:55) Martin & Zalubas (1983) d Al (cid:55) (cid:55) (cid:55) Martin & Zalubas (1979) d Ti (cid:55) (cid:55) (cid:55) Saloman (2012) d Legend : : This work, (cid:55) : Not applicable, -: Unnecessary, ? : No references provided, CCCBDB: NIST Computational ChemistryComparison and Benchmark Database (Johnson 2018).a) via ExoMol ( http://exomol.com/ ). b) via HITRAN (Gordon et al. 2017). c) via NIST chemistry WebBook ( https://doi.org/10.18434/T4D303 ). d) via NIST Atomic Spectra Database (Kramida et al. 2018). e) via CCCBDB (Johnson 2018) f) Uses g = and g = as para-ortho degeneracy which is preferred over using g = / and g = / like Vidler & Tennyson (2000). g) The mostlikely reference of list of the references provided by NIST chemistry WebBook ( https://doi.org/10.18434/T4D303 ). h) Unclearreference for the second energy level. i) First level: improved theoretical value over the theoretical one of Huber & Herzberg (1979).MNRAS000 , 1–68 (2019) ucleation in AGB winds E2.3 Al-bearing species
This section contains a more complete overview of all Al-bearing species results in the comprehensive chemical nu-cleation model using the polymer nucleation description(Figs. E25, E26).
APPENDIX F: CHEMICAL NETWORK
This appendix lists all the used reactions with their reactionrate coefficient and the source of this data (Tab. F1). Thisis the comprehensive chemical network used in this paper.Subsets of this network are not explicitly listed, i.e. theclosed nucleation networks.
MNRAS , 1–68 (2019) J. Boulangier et al.
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Temperature (K) − − − − − D e n s i t y ( k g m − ) TiO − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
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Temperature (K) − − − − − D e n s i t y ( k g m − ) (TiO ) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
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Temperature (K) − − − − − D e n s i t y ( k g m − ) (TiO ) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
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Temperature (K) − − − − − D e n s i t y ( k g m − ) (TiO ) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
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Temperature (K) − − − − − D e n s i t y ( k g m − ) (TiO ) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
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Temperature (K) − − − − − D e n s i t y ( k g m − ) (TiO ) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
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Temperature (K) − − − − − D e n s i t y ( k g m − ) (TiO ) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
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Temperature (K) − − − − − D e n s i t y ( k g m − ) (TiO ) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
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Temperature (K) − − − − − D e n s i t y ( k g m − ) (TiO ) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
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Temperature (K) − − − − − D e n s i t y ( k g m − ) (TiO ) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r ) Figure E1.
Overview of the normalised mass density after one year of all TiO -clusters for a closed nucleation model using the monomernucleation description. MNRAS000
Overview of the normalised mass density after one year of all TiO -clusters for a closed nucleation model using the monomernucleation description. MNRAS000 , 1–68 (2019) ucleation in AGB winds
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Temperature (K) T i m e ( d ) TiO N u m b e r d e n s i t y ( m − )
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Temperature (K) T i m e ( d ) (TiO ) N u m b e r d e n s i t y ( m − )
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Temperature (K) T i m e ( d ) (TiO ) N u m b e r d e n s i t y ( m − )
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Temperature (K) T i m e ( d ) (TiO ) N u m b e r d e n s i t y ( m − )
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Temperature (K) T i m e ( d ) (TiO ) N u m b e r d e n s i t y ( m − )
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Temperature (K) T i m e ( d ) (TiO ) N u m b e r d e n s i t y ( m − )
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Temperature (K) T i m e ( d ) (TiO ) N u m b e r d e n s i t y ( m − )
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Temperature (K) T i m e ( d ) (TiO ) N u m b e r d e n s i t y ( m − )
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Temperature (K) T i m e ( d ) (TiO ) N u m b e r d e n s i t y ( m − )
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Temperature (K) T i m e ( d ) (TiO ) N u m b e r d e n s i t y ( m − ) Figure E2.
Temporal evolution of the absolute number density of all TiO -clusters at the benchmark total gas density ρ = · − kg m − for a closed nucleation model using the monomer nucleation description.MNRAS , 1–68 (2019) J. Boulangier et al.
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Temperature (K) − − − − − D e n s i t y ( k g m − ) MgO − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
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Temperature (K) − − − − − D e n s i t y ( k g m − ) (MgO) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
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Temperature (K) − − − − − D e n s i t y ( k g m − ) (MgO) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
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Temperature (K) − − − − − D e n s i t y ( k g m − ) (MgO) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
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Temperature (K) − − − − − D e n s i t y ( k g m − ) (MgO) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
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Temperature (K) − − − − − D e n s i t y ( k g m − ) (MgO) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
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Temperature (K) − − − − − D e n s i t y ( k g m − ) (MgO) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
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Temperature (K) − − − − − D e n s i t y ( k g m − ) (MgO) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
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Temperature (K) − − − − − D e n s i t y ( k g m − ) (MgO) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
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Temperature (K) − − − − − D e n s i t y ( k g m − ) (MgO) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r ) Figure E3.
Overview of the normalised mass density after one year of all MgO-clusters for a closed nucleation model using the monomernucleation description. MNRAS000
Overview of the normalised mass density after one year of all MgO-clusters for a closed nucleation model using the monomernucleation description. MNRAS000 , 1–68 (2019) ucleation in AGB winds
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Temperature (K) T i m e ( d ) MgO N u m b e r d e n s i t y ( m − )
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Temperature (K) T i m e ( d ) (MgO) N u m b e r d e n s i t y ( m − )
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Temperature (K) T i m e ( d ) (MgO) N u m b e r d e n s i t y ( m − )
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Temperature (K) T i m e ( d ) (MgO) N u m b e r d e n s i t y ( m − )
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Temperature (K) T i m e ( d ) (MgO) N u m b e r d e n s i t y ( m − )
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Temperature (K) T i m e ( d ) (MgO) N u m b e r d e n s i t y ( m − )
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Temperature (K) T i m e ( d ) (MgO) N u m b e r d e n s i t y ( m − )
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Temperature (K) T i m e ( d ) (MgO) N u m b e r d e n s i t y ( m − )
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Temperature (K) T i m e ( d ) (MgO) N u m b e r d e n s i t y ( m − )
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Temperature (K) T i m e ( d ) (MgO) N u m b e r d e n s i t y ( m − ) Figure E4.
Temporal evolution of the absolute number density of all MgO-clusters at the benchmark total gas density ρ = · − kg m − for a closed nucleation model using the monomer nucleation description.MNRAS , 1–68 (2019) J. Boulangier et al.
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Temperature (K) T i m e ( h ) MgO N u m b e r d e n s i t y ( m − )
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Temperature (K) T i m e ( h ) (MgO) N u m b e r d e n s i t y ( m − )
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Temperature (K) T i m e ( h ) (MgO) N u m b e r d e n s i t y ( m − )
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Temperature (K) T i m e ( h ) (MgO) N u m b e r d e n s i t y ( m − )
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Temperature (K) T i m e ( h ) (MgO) N u m b e r d e n s i t y ( m − )
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Temperature (K) T i m e ( h ) (MgO) N u m b e r d e n s i t y ( m − )
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Temperature (K) T i m e ( h ) (MgO) N u m b e r d e n s i t y ( m − )
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Temperature (K) T i m e ( h ) (MgO) N u m b e r d e n s i t y ( m − )
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Temperature (K) T i m e ( h ) (MgO) N u m b e r d e n s i t y ( m − )
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Temperature (K) T i m e ( h ) (MgO) N u m b e r d e n s i t y ( m − ) Figure E5.
Refined temporal evolution of the absolute number density of all MgO-clusters at the benchmark total gas density ρ = · − kg m − for a closed nucleation model using the monomer nucleation description. MNRAS000
Refined temporal evolution of the absolute number density of all MgO-clusters at the benchmark total gas density ρ = · − kg m − for a closed nucleation model using the monomer nucleation description. MNRAS000 , 1–68 (2019) ucleation in AGB winds
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Temperature (K) − − − − − D e n s i t y ( k g m − ) SiO − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
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Temperature (K) − − − − − D e n s i t y ( k g m − ) (SiO) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
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Temperature (K) − − − − − D e n s i t y ( k g m − ) (SiO) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
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Temperature (K) − − − − − D e n s i t y ( k g m − ) (SiO) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
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Temperature (K) − − − − − D e n s i t y ( k g m − ) (SiO) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
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Temperature (K) − − − − − D e n s i t y ( k g m − ) (SiO) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
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Temperature (K) − − − − − D e n s i t y ( k g m − ) (SiO) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
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Temperature (K) − − − − − D e n s i t y ( k g m − ) (SiO) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
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Temperature (K) − − − − − D e n s i t y ( k g m − ) (SiO) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (SiO) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r ) Figure E6.
Overview of the normalised mass density after one year of all SiO-clusters for a closed nucleation model using the monomernucleation description.MNRAS , 1–68 (2019) J. Boulangier et al.
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) Al O − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (Al O ) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (Al O ) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (Al O ) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (Al O ) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (Al O ) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (Al O ) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (Al O ) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r ) Figure E7.
Overview of the normalised mass density after one year of all Al O -clusters for a closed nucleation model using the monomernucleation description. MNRAS000
Overview of the normalised mass density after one year of all Al O -clusters for a closed nucleation model using the monomernucleation description. MNRAS000 , 1–68 (2019) ucleation in AGB winds
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) Al O N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) (Al O ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) (Al O ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) (Al O ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) (Al O ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) (Al O ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) (Al O ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) (Al O ) N u m b e r d e n s i t y ( m − ) Figure E8.
Temporal evolution of the absolute number density of all Al O -clusters at the benchmark total gas density ρ = · − kg m − for a closed nucleation model using the monomer nucleation description.MNRAS , 1–68 (2019) J. Boulangier et al.
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( h ) Al O N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( h ) (Al O ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( h ) (Al O ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( h ) (Al O ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( h ) (Al O ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( h ) (Al O ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( h ) (Al O ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( h ) (Al O ) N u m b e r d e n s i t y ( m − ) Figure E9.
Refined temporal evolution of the absolute number density of all Al O -clusters at the benchmark total gas density ρ = · − kg m − for a closed nucleation model using the monomer nucleation description. MNRAS000
Refined temporal evolution of the absolute number density of all Al O -clusters at the benchmark total gas density ρ = · − kg m − for a closed nucleation model using the monomer nucleation description. MNRAS000 , 1–68 (2019) ucleation in AGB winds
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) TiO − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (TiO ) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (TiO ) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (TiO ) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (TiO ) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (TiO ) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (TiO ) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (TiO ) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (TiO ) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (TiO ) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r ) Figure E10.
Overview of the normalised mass density after one year of all TiO -clusters for a closed nucleation model using the polymernucleation description.MNRAS , 1–68 (2019) J. Boulangier et al.
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) TiO N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) (TiO ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) (TiO ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) (TiO ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) (TiO ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) (TiO ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) (TiO ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) (TiO ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) (TiO ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) (TiO ) N u m b e r d e n s i t y ( m − ) Figure E11.
Temporal evolution of the absolute number density of all TiO -clusters at the benchmark total gas density ρ = · − kg m − for a closed nucleation model using the polymer nucleation description. MNRAS000
Temporal evolution of the absolute number density of all TiO -clusters at the benchmark total gas density ρ = · − kg m − for a closed nucleation model using the polymer nucleation description. MNRAS000 , 1–68 (2019) ucleation in AGB winds
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) MgO − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (MgO) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (MgO) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (MgO) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (MgO) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (MgO) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (MgO) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (MgO) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (MgO) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (MgO) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r ) Figure E12.
Overview of the normalised mass density after one year of all MgO-clusters for a closed nucleation model using the polymernucleation description.MNRAS , 1–68 (2019) J. Boulangier et al.
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) MgO N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) (MgO) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) (MgO) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) (MgO) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) (MgO) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) (MgO) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) (MgO) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) (MgO) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) (MgO) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) (MgO) N u m b e r d e n s i t y ( m − ) Figure E13.
Temporal evolution of the absolute number density of all MgO-clusters at the benchmark total gas density ρ = · − kg m − for a closed nucleation model using the polymer nucleation description. MNRAS000
Temporal evolution of the absolute number density of all MgO-clusters at the benchmark total gas density ρ = · − kg m − for a closed nucleation model using the polymer nucleation description. MNRAS000 , 1–68 (2019) ucleation in AGB winds
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( h ) MgO N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( h ) (MgO) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( h ) (MgO) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( h ) (MgO) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( h ) (MgO) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( h ) (MgO) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( h ) (MgO) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( h ) (MgO) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( h ) (MgO) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( h ) (MgO) N u m b e r d e n s i t y ( m − ) Figure E14.
Refined temporal evolution of the absolute number density of all MgO-clusters at the benchmark total gas density ρ = · − kg m − for a closed nucleation model using the polymer nucleation description.MNRAS , 1–68 (2019) J. Boulangier et al.
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) SiO − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (SiO) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (SiO) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (SiO) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (SiO) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (SiO) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (SiO) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (SiO) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (SiO) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (SiO) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r ) Figure E15.
Overview of the normalised mass density after one year of all SiO-clusters for a closed nucleation model using the polymernucleation description. MNRAS000
Overview of the normalised mass density after one year of all SiO-clusters for a closed nucleation model using the polymernucleation description. MNRAS000 , 1–68 (2019) ucleation in AGB winds
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) Al O − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (Al O ) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (Al O ) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (Al O ) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (Al O ) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (Al O ) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (Al O ) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (Al O ) − − − − N o r m a li s e d m a ss d e n s i t y ( w . r . t . i n i t i a l m o n o m e r ) Figure E16.
Overview of the normalised mass density after one year of all Al O -clusters for a closed nucleation model using thepolymer nucleation description.MNRAS , 1–68 (2019) J. Boulangier et al.
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) Al O N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) (Al O ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) (Al O ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) (Al O ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) (Al O ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) (Al O ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) (Al O ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( d ) (Al O ) N u m b e r d e n s i t y ( m − ) Figure E17.
Temporal evolution of the absolute number density of all Al O -clusters at the benchmark total gas density ρ = · − kg m − for a closed nucleation model using the polymer nucleation description. MNRAS000
Temporal evolution of the absolute number density of all Al O -clusters at the benchmark total gas density ρ = · − kg m − for a closed nucleation model using the polymer nucleation description. MNRAS000 , 1–68 (2019) ucleation in AGB winds
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( h ) Al O N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( h ) (Al O ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( h ) (Al O ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( h ) (Al O ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( h ) (Al O ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( h ) (Al O ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( h ) (Al O ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500 3000
Temperature (K) T i m e ( h ) (Al O ) N u m b e r d e n s i t y ( m − ) Figure E18.
Refined temporal evolution of the absolute number density of all Al O -clusters at the benchmark total gas density ρ = · − kg m − for a closed nucleation model using the polymer nucleation description.MNRAS , 1–68 (2019) J. Boulangier et al.
600 800 1000 1200 T ( K )10 − − − − ( n i + / n i ) / ( n / n ) (TiO ) / TiO (TiO ) / (TiO ) (TiO ) / (TiO ) (TiO ) / (TiO ) (TiO ) / (TiO ) (TiO ) / (TiO ) (TiO ) / (TiO ) (TiO ) / (TiO ) Equilibrium
Figure E19.
The relative ratios of TiO -clusters do not reach the equilibrium ratios in the entire temperature range. At the highesttemperatures at which the nucleation is feasible, the model results (full lines) correspond to the equilibrium ratios (dotted line). At lowertemperatures, the clusters need more time to reach the equilibrium ratios since the interaction probability is lower. This continuousevolution is also visible in Fig. E11. The results are of the closed polymer nucleation model for the benchmark total gas density ρ = · − kg m − at the final time step (one year). The figure shows the ratios of two clusters w.r.t. the ratio of both largest clusters. Ifthe number density of any of the four clusters is below the numerical solver accuracy of · − cm − , the ratios are not shown.
600 800 1000 1200 1400 1600 T ( K )10 − − ( n i + / n i ) / ( n / n ) (MgO) / MgO(MgO) / (MgO) (MgO) / (MgO) (MgO) / (MgO) (MgO) / (MgO) (MgO) / (MgO) (MgO) / (MgO) (MgO) / (MgO) Equilibrium
Figure E20.
The relative ratios of MgO-clusters do not reach the equilibrium ratios in the entire temperature range. At the highesttemperatures at which the nucleation is feasible, the model results (full lines) correspond to the equilibrium ratios (dotted line). At lowertemperatures, the clusters need more time to reach the equilibrium ratios since the interaction probability is lower. This transition isvisible between to . The continuous evolution is also visible in Fig. E13. The results are of the closed polymer nucleation modelfor the benchmark total gas density ρ = · − kg m − at the final time step (one year). The figure shows the ratios of two clusters w.r.t.the ratio of both largest clusters. If the number density of any of the four clusters is below the numerical solver accuracy of · − cm − ,the ratios are not shown. Note that no model results involving (MgO) are visible since this cluster does not exists under the localconditions. MNRAS000
The relative ratios of MgO-clusters do not reach the equilibrium ratios in the entire temperature range. At the highesttemperatures at which the nucleation is feasible, the model results (full lines) correspond to the equilibrium ratios (dotted line). At lowertemperatures, the clusters need more time to reach the equilibrium ratios since the interaction probability is lower. This transition isvisible between to . The continuous evolution is also visible in Fig. E13. The results are of the closed polymer nucleation modelfor the benchmark total gas density ρ = · − kg m − at the final time step (one year). The figure shows the ratios of two clusters w.r.t.the ratio of both largest clusters. If the number density of any of the four clusters is below the numerical solver accuracy of · − cm − ,the ratios are not shown. Note that no model results involving (MgO) are visible since this cluster does not exists under the localconditions. MNRAS000 , 1–68 (2019) ucleation in AGB winds
500 1000 1500 2000 T ( K )10 − − ( n i + / n i ) / ( n / n ) (Al O ) / Al O (Al O ) / (Al O ) (Al O ) / (Al O ) (Al O ) / (Al O ) (Al O ) / (Al O ) (Al O ) / (Al O ) Equilibrium
Figure E21.
The relative ratios of Al O -clusters (full lines) do not correspond to the equilibrium ratios (dotted line). The figure showsthe ratios of two clusters w.r.t. the ratio of both largest clusters. If the number density of any of the four clusters is below the numericalsolver accuracy of · − cm − , the ratios are not shown. Due to the large variation in number densities of the clusters in differenttemperature regimes (order of magnitude), it is often impossible to compare ratios of the clusters. This variation is more clearly visiblein Fig. E17. The results are of the closed polymer nucleation model for the benchmark total gas density ρ = · − kg m − at the finaltime step (one year).MNRAS , 1–68 (2019) J. Boulangier et al.
500 1000 1500 2000 2500
Temperature (K) − − − − − D e n s i t y ( k g m − ) TiO − − − − N o r m a li s e d m e t a l m a ss d e n s i t y ( w . r . t . i n i t i a l a t o m i c m e t a l )
500 1000 1500 2000 2500
Temperature (K) − − − − − D e n s i t y ( k g m − ) (TiO ) − − − − N o r m a li s e d m e t a l m a ss d e n s i t y ( w . r . t . i n i t i a l a t o m i c m e t a l )
500 1000 1500 2000 2500
Temperature (K) − − − − − D e n s i t y ( k g m − ) (TiO ) − − − − N o r m a li s e d m e t a l m a ss d e n s i t y ( w . r . t . i n i t i a l a t o m i c m e t a l )
500 1000 1500 2000 2500
Temperature (K) − − − − − D e n s i t y ( k g m − ) (TiO ) − − − − N o r m a li s e d m e t a l m a ss d e n s i t y ( w . r . t . i n i t i a l a t o m i c m e t a l )
500 1000 1500 2000 2500
Temperature (K) − − − − − D e n s i t y ( k g m − ) (TiO ) − − − − N o r m a li s e d m e t a l m a ss d e n s i t y ( w . r . t . i n i t i a l a t o m i c m e t a l )
500 1000 1500 2000 2500
Temperature (K) − − − − − D e n s i t y ( k g m − ) (TiO ) − − − − N o r m a li s e d m e t a l m a ss d e n s i t y ( w . r . t . i n i t i a l a t o m i c m e t a l )
500 1000 1500 2000 2500
Temperature (K) − − − − − D e n s i t y ( k g m − ) (TiO ) − − − − N o r m a li s e d m e t a l m a ss d e n s i t y ( w . r . t . i n i t i a l a t o m i c m e t a l )
500 1000 1500 2000 2500
Temperature (K) − − − − − D e n s i t y ( k g m − ) (TiO ) − − − − N o r m a li s e d m e t a l m a ss d e n s i t y ( w . r . t . i n i t i a l a t o m i c m e t a l )
500 1000 1500 2000 2500
Temperature (K) − − − − − D e n s i t y ( k g m − ) (TiO ) − − − − N o r m a li s e d m e t a l m a ss d e n s i t y ( w . r . t . i n i t i a l a t o m i c m e t a l )
500 1000 1500 2000 2500
Temperature (K) − − − − − D e n s i t y ( k g m − ) (TiO ) − − − − N o r m a li s e d m e t a l m a ss d e n s i t y ( w . r . t . i n i t i a l a t o m i c m e t a l )
500 1000 1500 2000 2500
Temperature (K) − − − − − D e n s i t y ( k g m − ) Ti − − − − N o r m a li s e d m e t a l m a ss d e n s i t y ( w . r . t . i n i t i a l a t o m i c m e t a l )
500 1000 1500 2000 2500
Temperature (K) − − − − − D e n s i t y ( k g m − ) TiO − − − − N o r m a li s e d m e t a l m a ss d e n s i t y ( w . r . t . i n i t i a l a t o m i c m e t a l ) Figure E22.
Overview of the normalised mass density after one year of all Ti-bearing species for the comprehensive chemical nucleationmodel using the polymer nucleation description. MNRAS000
Overview of the normalised mass density after one year of all Ti-bearing species for the comprehensive chemical nucleationmodel using the polymer nucleation description. MNRAS000 , 1–68 (2019) ucleation in AGB winds
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Temperature (K) T i m e ( d ) TiO N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500
Temperature (K) T i m e ( d ) (TiO ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500
Temperature (K) T i m e ( d ) (TiO ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500
Temperature (K) T i m e ( d ) (TiO ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500
Temperature (K) T i m e ( d ) (TiO ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500
Temperature (K) T i m e ( d ) (TiO ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500
Temperature (K) T i m e ( d ) (TiO ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500
Temperature (K) T i m e ( d ) (TiO ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500
Temperature (K) T i m e ( d ) (TiO ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500
Temperature (K) T i m e ( d ) (TiO ) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500
Temperature (K) T i m e ( d ) Ti N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500
Temperature (K) T i m e ( d ) TiO N u m b e r d e n s i t y ( m − ) Figure E23.
Temporal evolution of the absolute number density of all Ti-bearing species at the benchmark total gas density ρ = · − kg m − for the comprehensive chemical nucleation model using the polymer nucleation description.MNRAS , 1–68 (2019) J. Boulangier et al.
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Temperature (K) − − − − − D e n s i t y ( k g m − ) SiO − − − − N o r m a li s e d m e t a l m a ss d e n s i t y ( w . r . t . i n i t i a l a t o m i c m e t a l )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (SiO) − − − − N o r m a li s e d m e t a l m a ss d e n s i t y ( w . r . t . i n i t i a l a t o m i c m e t a l )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (SiO) − − − − N o r m a li s e d m e t a l m a ss d e n s i t y ( w . r . t . i n i t i a l a t o m i c m e t a l )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (SiO) − − − − N o r m a li s e d m e t a l m a ss d e n s i t y ( w . r . t . i n i t i a l a t o m i c m e t a l )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (SiO) − − − − N o r m a li s e d m e t a l m a ss d e n s i t y ( w . r . t . i n i t i a l a t o m i c m e t a l )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (SiO) − − − − N o r m a li s e d m e t a l m a ss d e n s i t y ( w . r . t . i n i t i a l a t o m i c m e t a l )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (SiO) − − − − N o r m a li s e d m e t a l m a ss d e n s i t y ( w . r . t . i n i t i a l a t o m i c m e t a l )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (SiO) − − − − N o r m a li s e d m e t a l m a ss d e n s i t y ( w . r . t . i n i t i a l a t o m i c m e t a l )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (SiO) − − − − N o r m a li s e d m e t a l m a ss d e n s i t y ( w . r . t . i n i t i a l a t o m i c m e t a l )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) (SiO) − − − − N o r m a li s e d m e t a l m a ss d e n s i t y ( w . r . t . i n i t i a l a t o m i c m e t a l )
500 1000 1500 2000 2500 3000
Temperature (K) − − − − − D e n s i t y ( k g m − ) Si − − − − N o r m a li s e d m e t a l m a ss d e n s i t y ( w . r . t . i n i t i a l a t o m i c m e t a l ) Figure E24.
Overview of the normalised mass density of all Si-bearing species for the comprehensive chemical nucleation model usingthe polymer nucleation description. Species with zero abundance are not shown. MNRAS000
Overview of the normalised mass density of all Si-bearing species for the comprehensive chemical nucleation model usingthe polymer nucleation description. Species with zero abundance are not shown. MNRAS000 , 1–68 (2019) ucleation in AGB winds
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Temperature (K) − − − − − D e n s i t y ( k g m − ) Al N u m b e r d e n s i t y ( m − )
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Temperature (K) − − − − − D e n s i t y ( k g m − ) AlO N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500
Temperature (K) − − − − − D e n s i t y ( k g m − ) AlH N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500
Temperature (K) − − − − − D e n s i t y ( k g m − ) Al(OH) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500
Temperature (K) − − − − − D e n s i t y ( k g m − ) Al(OH) N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500
Temperature (K) − − − − − D e n s i t y ( k g m − ) AlC N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500
Temperature (K) − − − − − D e n s i t y ( k g m − ) AlOH N u m b e r d e n s i t y ( m − )
500 1000 1500 2000 2500
Temperature (K) − − − − − D e n s i t y ( k g m − ) AlO N u m b e r d e n s i t y ( m − ) Figure E25.
Overview of the absolute number density after one year of all Al-bearing species for the comprehensive chemical nucleationmodel using the polymer nucleation description. Species with zero abundance are not shown.
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Temperature (K) T i m e ( d ) Al . . . . . . N o r m a li s e dnu m b e r d e n s i t y ( w . r . t . i n i t i a l a t o m i c m e t a l )
500 1000 1500 2000 2500
Temperature (K) T i m e ( d ) AlO − − − − N o r m a li s e dnu m b e r d e n s i t y ( w . r . t . i n i t i a l a t o m i c m e t a l ) Figure E26.
Temporal evolution of the absolute number density of all Al-bearing species at the benchmark total gas density ρ = · − kg m − for the comprehensive chemical nucleation model using the polymer nucleation description. Species with zero abundanceare not shown.MNRAS , 1–68 (2019) J. Boulangier et al.
Table F1: An overview of the comprehensive chemical network. The chemical reactions are listed alphabetically. Additionalinformation on references, parameters, abbreviations, and notes are provided at the bottom of the table.No. Reaction Rate coefficient (cm ( N − ) s − ) with N number of reactants Ref.1 Al + AlO + M Al O + M k = . · − n tot exp (cid:16) − . T (cid:17) EQR (cid:0) T , (cid:8) Al , AlO (cid:9) , (cid:8) Al O (cid:9)(cid:1)
12 Al + C + M AlC + M k = . · − T . n tot
23 Al + CO AlO + CO k = . · − T . exp (cid:16) − T (cid:17)
24 Al + H + M AlH + M k = . · − T . n tot
35 Al + H AlH + H k = . · − T . EQR (cid:0) T , (cid:8) Al , H (cid:9) , { AlH , H } (cid:1)
36 Al + H O AlOH + H k = . · − exp (cid:16) − T (cid:17) T − . + . · − T . exp (cid:16) − T (cid:17)
47 Al + H O + H O AlO H + H k = . · − T .
38 Al + H O + H O AlOH + H + H O k = . · − T . exp (cid:16) − T (cid:17)
39 Al + H O AlOH + OH k = . · − T . exp (cid:16) . T (cid:17)
510 Al + H O AlO H + γ k = . · − T . exp (cid:16) . T (cid:17)
511 Al + HCO AlH + CO k = . · − T .
212 Al + HO AlO + OH k = . · − T .
213 Al + HO AlH + O k = . · − T .
214 Al + O + H AlO + H k = . · − T −
115 Al + O + H O AlO + H O k = . · − T −
116 Al + O + O AlO + O k = . · − T −
117 Al + O AlO + O k = . · − T .
218 Al + OH + M AlOH + M k = . · − T . n tot
219 Al O Al O + Al O k = k + N , M (cid:0) Al O , , , T (cid:1) EQR (cid:0) T , (cid:8) Al O (cid:9) , (cid:8) Al O , Al O (cid:9)(cid:1)
620 Al O Al O + Al O k = k + N , M (cid:0) Al O , , , T (cid:1) EQR (cid:0) T , (cid:8) Al O (cid:9) , (cid:8) Al O , Al O (cid:9)(cid:1)
621 Al O Al O + Al O k = k + N , M (cid:0) Al O , , , T (cid:1) EQR (cid:0) T , (cid:8) Al O (cid:9) , (cid:8) Al O , Al O (cid:9)(cid:1)
622 Al O Al O + Al O k = k + N , M (cid:0) Al O , , , T (cid:1) EQR (cid:0) T , (cid:8) Al O (cid:9) , (cid:8) Al O , Al O (cid:9)(cid:1)
623 Al O Al O + Al O k = k + N , M (cid:0) Al O , , , T (cid:1) EQR (cid:0) T , (cid:8) Al O (cid:9) , (cid:8) Al O (cid:9)(cid:1)
624 Al O Al O + Al O k = k + N , M (cid:0) Al O , , , T (cid:1) EQR (cid:0) T , (cid:8) Al O (cid:9) , (cid:8) Al O , Al O (cid:9)(cid:1)
625 Al O Al O + Al O k = k + N , M (cid:0) Al O , , , T (cid:1) EQR (cid:0) T , (cid:8) Al O (cid:9) , (cid:8) Al O , Al O (cid:9)(cid:1)
626 Al O Al O + Al O k = k + N , M (cid:0) Al O , , , T (cid:1) EQR (cid:0) T , (cid:8) Al O (cid:9) , (cid:8) Al O , Al O (cid:9)(cid:1)
627 Al O Al O + Al O k = k + N , M (cid:0) Al O , , , T (cid:1) EQR (cid:0) T , (cid:8) Al O (cid:9) , (cid:8) Al O , Al O (cid:9)(cid:1)
628 Al O Al O + Al O k = k + N , M (cid:0) Al O , , , T (cid:1) EQR (cid:0) T , (cid:8) Al O (cid:9) , (cid:8) Al O , Al O (cid:9)(cid:1)
629 Al O Al O + Al O k = k + N , M (cid:0) Al O , , , T (cid:1) EQR (cid:0) T , (cid:8) Al O (cid:9) , (cid:8) Al O , Al O (cid:9)(cid:1)
630 Al O Al O + Al O k = k + N , M (cid:0) Al O , , , T (cid:1) EQR (cid:0) T , (cid:8) Al O (cid:9) , (cid:8) Al O (cid:9)(cid:1)
631 Al O + M AlO + Al + M k = . · − n tot exp (cid:16) − . T (cid:17)
132 Al O + O + M Al O + M k = . · − n tot exp (cid:16) − T (cid:17) EQR (cid:0) T , (cid:8) Al O , O (cid:9) , (cid:8) Al O (cid:9)(cid:1)
133 Al O + M AlO + AlO + M k = . · − n tot exp (cid:16) − . T (cid:17)
134 Al O + M Al + AlO + M k = . · − n tot exp (cid:16) − . T (cid:17)
135 Al O + M Al O + O + M k = . · − n tot exp (cid:16) − T (cid:17)
136 Al O + O + M Al O + M k = . · − n tot exp (cid:16) − . T (cid:17) EQR (cid:0) T , (cid:8) Al O , O (cid:9) , (cid:8) Al O (cid:9)(cid:1)
137 Al O + M Al O + O + M k = . · − n tot exp (cid:16) − . T (cid:17) MNRAS000
137 Al O + M Al O + O + M k = . · − n tot exp (cid:16) − . T (cid:17) MNRAS000 , 1–68 (2019) ucleation in AGB winds No. Reaction Rate coefficient (cm ( N − ) s − ) with N number of reactants Ref.38 Al O + M AlO + AlO + M k = . · − n tot exp (cid:16) − . T (cid:17)
139 Al O + Al O Al O k = k + N , M (cid:0) Al O , , , T (cid:1)
640 Al O + Al O Al O k = k + N , M (cid:0) Al O , , , T (cid:1)
641 Al O + Al O Al O k = k + N , M (cid:0) Al O , , , T (cid:1)
642 Al O + Al O Al O k = k + N , M (cid:0) Al O , , , T (cid:1)
643 Al O + Al O Al O k = k + N , M (cid:0) Al O , , , T (cid:1)
644 Al O + Al O Al O k = k + N , M (cid:0) Al O , , , T (cid:1)
645 Al O + Al O Al O k = k + N , M (cid:0) Al O , , , T (cid:1)
646 Al O Al O + Al O k = k + N , M (cid:0) Al O , , , T (cid:1) EQR (cid:0) T , (cid:8) Al O (cid:9) , (cid:8) Al O (cid:9)(cid:1)
647 Al O + Al O Al O k = k + N , M (cid:0) Al O , , , T (cid:1)
648 Al O + Al O Al O k = k + N , M (cid:0) Al O , , , T (cid:1)
649 Al O + Al O Al O k = k + N , M (cid:0) Al O , , , T (cid:1)
650 Al O + Al O Al O k = k + N , M (cid:0) Al O , , , T (cid:1)
651 Al O + Al O Al O k = k + N , M (cid:0) Al O , , , T (cid:1)
652 Al O Al O + Al O k = k + N , M (cid:0) Al O , , , T (cid:1) EQR (cid:0) T , (cid:8) Al O (cid:9) , (cid:8) Al O , Al O (cid:9)(cid:1)
653 Al O + Al O Al O k = k + N , M (cid:0) Al O , , , T (cid:1)
654 Al O + Al O Al O k = k + N , M (cid:0) Al O , , , T (cid:1)
655 Al O + Al O Al O k = k + N , M (cid:0) Al O , , , T (cid:1)
656 Al O Al O + Al O k = k + N , M (cid:0) Al O , , , T (cid:1) EQR (cid:0) T , (cid:8) Al O (cid:9) , (cid:8) Al O , Al O (cid:9)(cid:1)
657 Al O Al O + Al O k = k + N , M (cid:0) Al O , , , T (cid:1) EQR (cid:0) T , (cid:8) Al O (cid:9) , (cid:8) Al O (cid:9)(cid:1)
658 Al O + Al O Al O k = k + N , M (cid:0) Al O , , , T (cid:1)
659 AlC + M Al + C + M k = . · − T . n tot EQR ( T , { AlC } , { Al , C })
260 AlH + M Al + H + M k = . · − T . n tot EQR ( T , { AlH } , { Al , H })
361 AlH + CO Al + HCO k = . · − T . EQR ( T , { AlH , CO } , { Al , HCO })
262 AlH + H Al + H k = . · − T .
363 AlH + H + M AlH + M k = k Troe (cid:16) . · − exp (cid:16) − T (cid:17) , . · − exp (cid:16) − T (cid:17) , exp (cid:16) − T (cid:17) − . (cid:16) − . T (cid:17) + . (cid:16) − . T (cid:17) , n tot (cid:17) · EQR (cid:16) T , (cid:110) AlH , H (cid:111) , (cid:110) AlH (cid:111)(cid:17)
764 AlH + H + M AlH + M k = k Troe (cid:16) . · − exp (cid:16) − T (cid:17) , . · − exp (cid:16) − T (cid:17) , exp (cid:16) − T (cid:17) + .
06 exp (cid:16) − . T (cid:17) + .
94 exp (cid:16) − . T (cid:17) , n tot (cid:17) · EQR (cid:16) T , (cid:110) AlH , H (cid:111) , (cid:110) AlH (cid:111)(cid:17)
765 AlH + H AlH + H k = . · − EQR (cid:0) T , (cid:8) AlH , H (cid:9) , (cid:8) AlH , H (cid:9)(cid:1)
766 AlH + HO AlOH + OH k = . · − T . exp (cid:16) . T (cid:17)
567 AlH + O Al + HO k = . · − T . EQR (cid:0) T , (cid:8) AlH , O (cid:9) , (cid:8) Al , HO (cid:9)(cid:1)
268 AlH + M AlH + H + M k = k Troe (cid:16) . · − exp (cid:16) − T (cid:17) , . · − exp (cid:16) − T (cid:17) , exp (cid:16) − T (cid:17) − . (cid:16) − . T (cid:17) + . (cid:16) − . T (cid:17) , n tot (cid:17)
769 AlH + H AlH + H k = . · −
770 AlH + H AlH + H k = . · − T . EQR (cid:0) T , (cid:8) AlH , H (cid:9) , (cid:8) AlH , H (cid:9)(cid:1) MNRAS , 1–68 (2019) J. Boulangier et al.
No. Reaction Rate coefficient (cm ( N − ) s − ) with N number of reactants Ref.71 AlH + M AlH + H + M k = k Troe (cid:16) . · − exp (cid:16) − T (cid:17) , . · − exp (cid:16) − T (cid:17) , exp (cid:16) − T (cid:17) + .
06 exp (cid:16) − . T (cid:17) + .
94 exp (cid:16) − . T (cid:17) , n tot (cid:17)
772 AlH + H AlH + H k = . · − T .
773 AlO + Al + M Al O + M k = . · − n tot exp (cid:16) − . T (cid:17) EQR (cid:0) T , { Al , AlO } , (cid:8) Al O (cid:9)(cid:1)
174 AlO + AlH AlOH + Al k = . · − T .
375 AlO + AlO + M Al O + M k = . · − n tot exp (cid:16) − . T (cid:17) EQR (cid:0) T , { AlO } , (cid:8) Al O (cid:9)(cid:1)
176 AlO + CO Al + CO k = . · − T . exp (cid:16) − T (cid:17) EQR (cid:0) T , { AlO , CO } , (cid:8) Al , CO (cid:9)(cid:1)
277 AlO + CO AlO + CO k = . · − T . exp (cid:16) − T (cid:17)
278 AlO + H + M AlOH + M k = . · − T . n tot
279 AlO + H AlOH + H k = . · − T . exp (cid:16) − T (cid:17) EQR (cid:0) T , (cid:8) AlO , H (cid:9) , { AlOH , H } (cid:1)
480 AlO + H Al + O + H k = . · − T − EQR ( T , { AlO } , { Al , O })
181 AlO + H O Al + O + H O k = . · − T − EQR ( T , { AlO } , { Al , O })
182 AlO + H O AlOH + HO k = . · − T . exp (cid:16) . T (cid:17)
583 AlO + HO AlOH + O k = . · − T − . exp (cid:16) T (cid:17)
284 AlO + O Al + O k = . · − T . EQR (cid:0) T , { AlO , O } , (cid:8) Al , O (cid:9)(cid:1)
285 AlO + O + M AlO + M k = . · − n tot exp (cid:16) − . T (cid:17) EQR (cid:0) T , { AlO , O } , (cid:8) AlO (cid:9)(cid:1)
186 AlO + O AlO + O k = . · − T . exp (cid:16) − T (cid:17)
287 AlO + O Al + O + O k = . · − T − EQR ( T , { AlO } , { Al , O })
188 AlO + OH + M AlO H + M k = . · − T . n tot
289 AlO + OH AlOH + O k = . · − T . exp (cid:16) − T (cid:17) EQR ( T , { AlO , OH } , { AlOH , O })
290 AlO + OH Al + HO k = . · − T . EQR (cid:0) T , { AlO , OH } , (cid:8) Al , HO (cid:9)(cid:1)
291 AlO + M AlO + O + M k = . · − n tot exp (cid:16) − . T (cid:17)
192 AlO + AlO + M Al O + M k = . · − n tot exp (cid:16) − . T (cid:17) EQR (cid:0) T , (cid:8) AlO , AlO (cid:9) , (cid:8) Al O (cid:9)(cid:1)
193 AlO + CO AlO + CO k = . · − T . exp (cid:16) − T (cid:17) EQR (cid:0) T , (cid:8) AlO , CO (cid:9) , (cid:8) AlO , CO (cid:9)(cid:1)
294 AlO + H + M AlO H + M k = . · − T . n tot
295 AlO + H AlO H + H k = . · − T . exp (cid:16) − T (cid:17)
296 AlO + H O AlO H + OH k = . · − T . exp (cid:16) − T (cid:17)
297 AlO + O AlO + O k = . · − T . exp (cid:16) − T (cid:17) EQR (cid:0) T , (cid:8) AlO , O (cid:9) , (cid:8) AlO , O (cid:9)(cid:1)
298 AlO + OH AlO H + O k = . · − T .
299 AlO H + M AlO + OH + M k = . · − T . n tot EQR (cid:0) T , (cid:8) AlO H (cid:9) , { AlO , OH } (cid:1) H + M AlO + H + M k = . · − T . n tot EQR (cid:0) T , (cid:8) AlO H (cid:9) , (cid:8) AlO , H (cid:9)(cid:1) H + M AlOH + O + M k = . · − T . n tot EQR (cid:0) T , (cid:8) AlO H (cid:9) , { AlOH , O } (cid:1) H + H AlO H + γ k = . · − T . H + H AlO + H k = . · − T . exp (cid:16) − T (cid:17) EQR (cid:0) T , (cid:8) AlO H , H (cid:9) , (cid:8) AlO , H (cid:9)(cid:1) H + H O AlOH + H O k = . · − T . exp (cid:16) . T (cid:17) EQR (cid:0) T , (cid:8) AlO H , H O (cid:9) , (cid:8) AlOH , H O (cid:9)(cid:1) H + O AlOH + O k = . · − T . H + O AlO + OH k = . · − T . EQR (cid:0) T , (cid:8) AlO H , O (cid:9) , (cid:8) AlO , OH (cid:9)(cid:1) H + OH AlO + H O k = . · − T . exp (cid:16) − T (cid:17) EQR (cid:0) T , (cid:8) AlO H , OH (cid:9) , (cid:8) AlO , H O (cid:9)(cid:1) H + OH AlOH + HO k = . · − T . EQR (cid:0) T , (cid:8) AlO H , OH (cid:9) , (cid:8) AlOH , HO (cid:9)(cid:1) MNRAS000
299 AlO H + M AlO + OH + M k = . · − T . n tot EQR (cid:0) T , (cid:8) AlO H (cid:9) , { AlO , OH } (cid:1) H + M AlO + H + M k = . · − T . n tot EQR (cid:0) T , (cid:8) AlO H (cid:9) , (cid:8) AlO , H (cid:9)(cid:1) H + M AlOH + O + M k = . · − T . n tot EQR (cid:0) T , (cid:8) AlO H (cid:9) , { AlOH , O } (cid:1) H + H AlO H + γ k = . · − T . H + H AlO + H k = . · − T . exp (cid:16) − T (cid:17) EQR (cid:0) T , (cid:8) AlO H , H (cid:9) , (cid:8) AlO , H (cid:9)(cid:1) H + H O AlOH + H O k = . · − T . exp (cid:16) . T (cid:17) EQR (cid:0) T , (cid:8) AlO H , H O (cid:9) , (cid:8) AlOH , H O (cid:9)(cid:1) H + O AlOH + O k = . · − T . H + O AlO + OH k = . · − T . EQR (cid:0) T , (cid:8) AlO H , O (cid:9) , (cid:8) AlO , OH (cid:9)(cid:1) H + OH AlO + H O k = . · − T . exp (cid:16) − T (cid:17) EQR (cid:0) T , (cid:8) AlO H , OH (cid:9) , (cid:8) AlO , H O (cid:9)(cid:1) H + OH AlOH + HO k = . · − T . EQR (cid:0) T , (cid:8) AlO H , OH (cid:9) , (cid:8) AlOH , HO (cid:9)(cid:1) MNRAS000 , 1–68 (2019) ucleation in AGB winds No. Reaction Rate coefficient (cm ( N − ) s − ) with N number of reactants Ref.109 AlO H AlO H + H k = . · − T . EQR (cid:0) T , (cid:8) AlO H (cid:9) , (cid:8) AlO H , H (cid:9)(cid:1) H AlOH + OH k = . · − T . exp (cid:16) T (cid:17) EQR (cid:0) T , (cid:8) AlO H (cid:9) , { AlOH , OH } (cid:1) H Al + H O k = . · − T . exp (cid:16) . T (cid:17) EQR (cid:0) T , (cid:8) AlO H (cid:9) , (cid:8) Al , H O (cid:9)(cid:1) H + H Al + H O + H O k = . · − T . EQR (cid:0) T , (cid:8) AlO H , H (cid:9) , (cid:8) Al , H O (cid:9)(cid:1) H + OH AlO H + γ k = . · − T . exp (cid:16) T (cid:17) H + OH AlOH + H O k = . · − T . exp (cid:16) . T (cid:17) EQR (cid:0) T , (cid:8) AlO H , OH (cid:9) , (cid:8) AlOH , H O (cid:9)(cid:1) H AlO H + OH k = . · − T . exp (cid:16) T (cid:17) EQR (cid:0) T , (cid:8) AlO H (cid:9) , (cid:8) AlO H , OH (cid:9)(cid:1) H AlOH + H O k = . · − T . exp (cid:16) . T (cid:17) EQR (cid:0) T , (cid:8) AlO H (cid:9) , (cid:8) AlOH , H O (cid:9)(cid:1) = . · − T . n tot EQR ( T , { AlOH } , { AlO , H }) = . · − T . n tot EQR ( T , { AlOH } , { Al , OH }) = . · − T . EQR ( T , { Al , AlOH } , { AlH , AlO }) k = . · − T . exp (cid:16) − T (cid:17) O k = (cid:104) T − . . · − exp (cid:16) − T (cid:17) + . · − T . exp (cid:16) − T (cid:17)(cid:105) · EQR (cid:16) T , (cid:110) AlOH , H (cid:111) , (cid:110) Al , H O (cid:111)(cid:17) O Al + H O + H O k = . · − T . exp (cid:16) − T (cid:17) EQR (cid:0) T , (cid:8) AlOH , H , H O (cid:9) , (cid:8) Al , H O (cid:9)(cid:1) O AlO H + OH k = . · − T . exp (cid:16) . T (cid:17) O AlO H + γ k = . · − T . exp (cid:16) . T (cid:17) O AlO H + H O k = . · − T . exp (cid:16) . T (cid:17) AlO H + OH k = . · − T . AlO + H O k = . · − T . exp (cid:16) . T (cid:17) EQR (cid:0) T , (cid:8) AlOH , HO (cid:9) , (cid:8) AlO , H O (cid:9)(cid:1) = . · − T . exp (cid:16) − T (cid:17) H + M k = . · − T . n tot AlO H + O k = . · − T . EQR (cid:0) T , (cid:8) AlOH , O (cid:9) , (cid:8) AlO H , O (cid:9)(cid:1) AlO + HO k = T − . . · − exp (cid:16) T (cid:17) EQR (cid:0) T , (cid:8) AlOH , O (cid:9) , (cid:8) AlO , HO (cid:9)(cid:1) H + γ k = . · − T . exp (cid:16) T (cid:17) O k = . · − T . exp (cid:16) . T (cid:17) EQR (cid:0) T , { AlOH , OH } , (cid:8) Al , H O (cid:9)(cid:1) k = . · − T . exp (cid:16) . T (cid:17) EQR (cid:0) T , { AlOH , OH } , (cid:8) AlH , HO (cid:9)(cid:1) + γ k = . · − ( T / ) . exp (cid:16) − . T (cid:17) + H k = . · − + N k = . · − exp (cid:16) − T (cid:17) + O k = . · − ( T / ) . exp (cid:16) − T (cid:17) CR C + + e – k = . ζ k = . · − ( T / ) . exp (cid:16) − T (cid:17) – C – + γ k = . · − + CO + CH + k = . · − = · − = . · − ( T / ) . exp (cid:16) − T (cid:17) γ k = . · − ( T / ) . exp (cid:16) − T (cid:17) MNRAS , 1–68 (2019) J. Boulangier et al.
No. Reaction Rate coefficient (cm ( N − ) s − ) with N number of reactants Ref.146 C + N CN + N k = . · − exp (cid:16) − T (cid:17) = . · − ( T / ) . exp (cid:16) − T (cid:17) = . · − = · − ( T / ) − . = · − ( T / ) − . = . · − ( T / ) − . γ k = . · − ( T / ) . exp (cid:16) . T (cid:17) CO + O k = . · − ( T / ) . exp (cid:16) . T (cid:17) = . · − ( T / ) . exp (cid:16) − T (cid:17) = · − γ k = . · − ( T / ) . = . · − = . · − CO + SO k = · − + Si + + CO k = · − + + e – C + γ k = . · − ( T / ) − . exp (cid:16) . T (cid:17) + + Fe Fe + + C k = . · − + + Mg Mg + + C k = . · − + + Si Si + + C k = . · − – + H + C + H k = . · − ( T / ) − . + S CS + C k = · − = . · − ( T / ) . exp (cid:16) − T (cid:17) = . · − ( T / ) − . = . · − exp (cid:16) − T (cid:17) = . · − ( T / ) . exp (cid:16) . T (cid:17) + + e – k = . · − ( T / ) − . exp (cid:16) − . T (cid:17) = · − = . · − ( T / ) . exp (cid:16) − T (cid:17) + + e – C + H k = . · − ( T / ) − . OCN + O k = . · − ( T / ) − . exp (cid:16) . T (cid:17) = . · − ( T / ) . exp (cid:16) − T (cid:17) CR CO + + e – k = . ζ CR C + O k = ζ + + e – O + C k = · − ( T / ) − . + + e – F + γ k = . · − (cid:16) T (cid:17) . + + e – Fe + γ k = . · − ( T / ) − . γ k = · − – CH + e – k = · − CH + C k = . · − ( T / ) . exp (cid:16) − T (cid:17) MNRAS000
No. Reaction Rate coefficient (cm ( N − ) s − ) with N number of reactants Ref.146 C + N CN + N k = . · − exp (cid:16) − T (cid:17) = . · − ( T / ) . exp (cid:16) − T (cid:17) = . · − = · − ( T / ) − . = · − ( T / ) − . = . · − ( T / ) − . γ k = . · − ( T / ) . exp (cid:16) . T (cid:17) CO + O k = . · − ( T / ) . exp (cid:16) . T (cid:17) = . · − ( T / ) . exp (cid:16) − T (cid:17) = · − γ k = . · − ( T / ) . = . · − = . · − CO + SO k = · − + Si + + CO k = · − + + e – C + γ k = . · − ( T / ) − . exp (cid:16) . T (cid:17) + + Fe Fe + + C k = . · − + + Mg Mg + + C k = . · − + + Si Si + + C k = . · − – + H + C + H k = . · − ( T / ) − . + S CS + C k = · − = . · − ( T / ) . exp (cid:16) − T (cid:17) = . · − ( T / ) − . = . · − exp (cid:16) − T (cid:17) = . · − ( T / ) . exp (cid:16) . T (cid:17) + + e – k = . · − ( T / ) − . exp (cid:16) − . T (cid:17) = · − = . · − ( T / ) . exp (cid:16) − T (cid:17) + + e – C + H k = . · − ( T / ) − . OCN + O k = . · − ( T / ) − . exp (cid:16) . T (cid:17) = . · − ( T / ) . exp (cid:16) − T (cid:17) CR CO + + e – k = . ζ CR C + O k = ζ + + e – O + C k = · − ( T / ) − . + + e – F + γ k = . · − (cid:16) T (cid:17) . + + e – Fe + γ k = . · − ( T / ) − . γ k = · − – CH + e – k = · − CH + C k = . · − ( T / ) . exp (cid:16) − T (cid:17) MNRAS000 , 1–68 (2019) ucleation in AGB winds No. Reaction Rate coefficient (cm ( N − ) s − ) with N number of reactants Ref.185 H + CH C + H k = . · − exp (cid:16) − T (cid:17) = · − exp (cid:16) − T (cid:17) + C + + H k = . · − ( T / ) − . exp (cid:16) − . T (cid:17) CH + H k = . · − = . · − ( T / ) . exp (cid:16) − T (cid:17) CO + OH k = . · − exp (cid:16) − T (cid:17) CR H + + e – k = . ζ – H – + γ k = . · − ( T / ) . exp (cid:16) − . T (cid:17) – H + + e – + e – k = exp (cid:16) − . · − ln (cid:16) T e (cid:17) + . (cid:16) T e (cid:17) − . (cid:16) T e (cid:17) + . (cid:16) T e (cid:17) − . (cid:16) T e (cid:17) + . (cid:16) T e (cid:17) − . (cid:16) T e (cid:17) + . (cid:16) T e (cid:17) − . (cid:17) + H k = · − T − . + · − T − . + He k = . · − T − . H + H + H k = . · − ( T / ) − exp (cid:16) − T (cid:17) +2 H + H + k = . · − O OH + H + H k = . · − exp (cid:16) − T (cid:17) O OH + H k = . · − ( T / ) . exp (cid:16) − T (cid:17) S HS + H k = . · − ( T / ) . exp (cid:16) − T (cid:17) k = . · − exp (cid:16) − T (cid:17) k = . · − k = . · − + S + + H k = . · − + He + H + k = . · − ( T / ) . + He + H +2 k = . · − k = . · − ( T / ) . exp (cid:16) − T (cid:17) NH + H k = . · − ( T / ) . exp (cid:16) − T (cid:17) = . · − exp (cid:16) − T (cid:17) = . · − ( T / ) − . exp (cid:16) − T (cid:17) = . · − ( T / ) . exp (cid:16) − T (cid:17) = . · − ( T / ) . exp (cid:16) − T (cid:17) γ k = . · − ( T / ) − . + O + H + k = . · − ( T / ) . exp (cid:16) . T (cid:17) – OH + e – k = · − O + O + H k = · − exp (cid:16) − T (cid:17) OH + O k = . · − exp (cid:16) − T (cid:17) OH + O k = . · − exp (cid:16) T (cid:17) MNRAS , 1–68 (2019) J. Boulangier et al.
No. Reaction Rate coefficient (cm ( N − ) s − ) with N number of reactants Ref.219 H + OCN OH + CN k = · − = . · − ( T / ) . exp (cid:16) − T (cid:17) = . · − exp (cid:16) − T (cid:17) = . · − exp (cid:16) − T (cid:17) = · − exp (cid:16) − T (cid:17) k = . · − ( T / ) . exp (cid:16) − T (cid:17) – HS + e – k = · − HS + S k = . · − ( T / ) . exp (cid:16) − T (cid:17) = . · − ( T / ) − . exp (cid:16) − T (cid:17) = . · − ( T / ) . exp (cid:16) − T (cid:17) + SiH + + γ k = . · − ( T / ) − . + Si + + H k = . · − + + e – H + γ k = . · − ( T / ) − . + + Fe Fe + + H k = . · − + + H H +2 + γ k = . · − ( T / ) . exp (cid:16) − T (cid:17) + + Mg Mg + + H k = . · − + + NH NH + + H k = . · − ( T / ) − . + + Na Na + + H k = . · − + + O O + + H k = . · − ( T / ) . exp (cid:16) − . T (cid:17) + + OH OH + + H k = . · − ( T / ) − . + + P P + + H k = · − + + S S + + H k = . · − + + Si Si + + H k = . · − + + SiO SiO + + H k = . · − ( T / ) − . – + C CH + e – k = · − – + e – H + e – + e – k = exp (cid:16) − . · − ln (cid:16) T e (cid:17) + . (cid:16) T e (cid:17) − . (cid:16) T e (cid:17) + . (cid:16) T e (cid:17) − . (cid:16) T e (cid:17) + . (cid:16) T e (cid:17) − . (cid:16) T e (cid:17) + . (cid:16) T e (cid:17) − . (cid:17) – + Fe + H + Fe k = . · − ( T / ) − . – + H H + e – k = . · − ( T / ) . exp (cid:16) − . T (cid:17) – + H H + H + e – k = exp (cid:16) − . · − ln (cid:16) T e (cid:17) + . · − ln (cid:16) T e (cid:17) − . · − ln (cid:16) T e (cid:17) + . · − ln (cid:16) T e (cid:17) + . (cid:16) T e (cid:17) − . (cid:16) T e (cid:17) + . (cid:16) T e (cid:17) − . (cid:16) T e (cid:17) + . (cid:16) T e (cid:17) − . (cid:17) MNRAS000
No. Reaction Rate coefficient (cm ( N − ) s − ) with N number of reactants Ref.219 H + OCN OH + CN k = · − = . · − ( T / ) . exp (cid:16) − T (cid:17) = . · − exp (cid:16) − T (cid:17) = . · − exp (cid:16) − T (cid:17) = · − exp (cid:16) − T (cid:17) k = . · − ( T / ) . exp (cid:16) − T (cid:17) – HS + e – k = · − HS + S k = . · − ( T / ) . exp (cid:16) − T (cid:17) = . · − ( T / ) − . exp (cid:16) − T (cid:17) = . · − ( T / ) . exp (cid:16) − T (cid:17) + SiH + + γ k = . · − ( T / ) − . + Si + + H k = . · − + + e – H + γ k = . · − ( T / ) − . + + Fe Fe + + H k = . · − + + H H +2 + γ k = . · − ( T / ) . exp (cid:16) − T (cid:17) + + Mg Mg + + H k = . · − + + NH NH + + H k = . · − ( T / ) − . + + Na Na + + H k = . · − + + O O + + H k = . · − ( T / ) . exp (cid:16) − . T (cid:17) + + OH OH + + H k = . · − ( T / ) − . + + P P + + H k = · − + + S S + + H k = . · − + + Si Si + + H k = . · − + + SiO SiO + + H k = . · − ( T / ) − . – + C CH + e – k = · − – + e – H + e – + e – k = exp (cid:16) − . · − ln (cid:16) T e (cid:17) + . (cid:16) T e (cid:17) − . (cid:16) T e (cid:17) + . (cid:16) T e (cid:17) − . (cid:16) T e (cid:17) + . (cid:16) T e (cid:17) − . (cid:16) T e (cid:17) + . (cid:16) T e (cid:17) − . (cid:17) – + Fe + H + Fe k = . · − ( T / ) − . – + H H + e – k = . · − ( T / ) . exp (cid:16) − . T (cid:17) – + H H + H + e – k = exp (cid:16) − . · − ln (cid:16) T e (cid:17) + . · − ln (cid:16) T e (cid:17) − . · − ln (cid:16) T e (cid:17) + . · − ln (cid:16) T e (cid:17) + . (cid:16) T e (cid:17) − . (cid:16) T e (cid:17) + . (cid:16) T e (cid:17) − . (cid:16) T e (cid:17) + . (cid:16) T e (cid:17) − . (cid:17) MNRAS000 , 1–68 (2019) ucleation in AGB winds No. Reaction Rate coefficient (cm ( N − ) s − ) with N number of reactants Ref.248 H – + H + H + H k = . · − ( T / ) − . – + H + H +2 + e – k = · − T − . – + Mg + H + Mg k = . · − ( T / ) − . – + N NH + e – k = · − – + Na + H + Na k = . · − ( T / ) − . – + O OH + e – k = · − – + O + H + O k = . · − ( T / ) − . – + S + H + S k = . · − ( T / ) − . – + Si + H + Si k = . · − ( T / ) − . + C CH + H k = . · − exp (cid:16) − T (cid:17) + C CH + γ k = · − + CH CH + H k = . · − exp (cid:16) − T (cid:17) + CN HCN + H k = . · − ( T / ) . exp (cid:16) − T (cid:17) CR H + + H – k = . ζ CR H + + H + e – k = . ζ CR H + H k = . ζ CR H +2 + e – k = . ζ + e – H + H + e – k = . · − ( T / ) . exp (cid:16) − T (cid:17) + e – H + H – k = . T − . exp (cid:16) − T (cid:17) + F HF + H k = · − exp (cid:16) − T (cid:17) + F + H +2 + F k = . · − + H + H H + H k = . · − T − . + T − . . · − + H + H +2 + H k = (cid:16) . · − ln (cid:16) T (cid:17) − . · − ln (cid:16) T (cid:17) + . · − ln (cid:16) T (cid:17) − . · − ln (cid:16) T (cid:17) + . · − ln (cid:16) T (cid:17) − . · − ln (cid:16) T (cid:17) + . · − ln (cid:16) T (cid:17) − . · − (cid:17) exp (cid:16) − . T (cid:17) + HS H S + H k = . · − ( T / ) . exp (cid:16) − T (cid:17) + N NH + H k = . · − exp (cid:16) − T (cid:17) + NH NH + H k = . · − exp (cid:16) − T (cid:17) + O OH + H k = . · − ( T / ) . exp (cid:16) − T (cid:17) + O + OH + + H k = . · − + O OH + OH k = . · − exp (cid:16) − T (cid:17) + OH H O + H k = . · − ( T / ) . exp (cid:16) − T (cid:17) + S HS + H k = . · − ( T / ) . exp (cid:16) − T (cid:17) + S + HS + + H k = . · − exp (cid:16) − T (cid:17) +2 + C CH + + H k = . · − +2 + e – H + H k = . · − ( T / ) − . +2 + He HeH + + H k = . · − +2 + O OH + + H k = . · − MNRAS , 1–68 (2019) J. Boulangier et al.
No. Reaction Rate coefficient (cm ( N − ) s − ) with N number of reactants Ref.284 HCO + + e – CO + H k = . · − ( T / ) − . + + Fe Fe + + HCO k = . · − + SiF + + H k = . · − ( T / ) − . S + S k = . · − + + e – S + H k = · − ( T / ) − . CR He + + e – k = . ζ – He + + e – + e – k = exp (cid:16) − . · − ln (cid:16) T e (cid:17) + . (cid:16) T e (cid:17) − . (cid:16) T e (cid:17) + . (cid:16) T e (cid:17) − . (cid:16) T e (cid:17) + . (cid:16) T e (cid:17) − . (cid:16) T e (cid:17) + . (cid:16) T e (cid:17) − . (cid:17) + + e – He + γ k = . · − ( T / ) − . + + HF F + + H + He k = . · − ( T / ) − . + + Si Si + + He k = . · − + + SiO O + Si + + He k = · − + + e – He + H k = · − ( T / ) − . – Mg + + e – + e – k = . · − (cid:16) T (cid:17) . O MgOH + H k = · − EQR (cid:0) T , (cid:8) H O , Mg (cid:9) , { H , MgOH } (cid:1) + HCO + Mg + k = . · − MgO + NO k = . · − exp (cid:16) − . R kJ T (cid:17) MgO + O k = . · − T . EQR (cid:0) T , (cid:8) MgO , O (cid:9) , { MgO , O } (cid:1) MgO + O k = . · − exp (cid:16) − T (cid:17) + S + Mg + k = . · − + Si + Mg + k = . · − + SiO + Mg + k = · − + + e – Mg + γ k = . · − ( T / ) − . O MgO + Mg O k = k + N , M ( MgO , , , T ) EQR (cid:0) T , (cid:8) Mg O (cid:9) , (cid:8) Mg O , MgO (cid:9)(cid:1) O Mg O + Mg O k = k + N , M ( MgO , , , T ) EQR (cid:0) T , (cid:8) Mg O (cid:9) , (cid:8) Mg O , Mg O (cid:9)(cid:1) O Mg O + Mg O k = k + N , M ( MgO , , , T ) EQR (cid:0) T , (cid:8) Mg O (cid:9) , (cid:8) Mg O , Mg O (cid:9)(cid:1) O Mg O + Mg O k = k + N , M ( MgO , , , T ) EQR (cid:0) T , (cid:8) Mg O (cid:9) , (cid:8) Mg O , Mg O (cid:9)(cid:1) O Mg O + Mg O k = k + N , M ( MgO , , , T ) EQR (cid:0) T , (cid:8) Mg O (cid:9) , (cid:8) Mg O (cid:9)(cid:1) O MgO + MgO k = k + N , M ( MgO , , , T ) EQR (cid:0) T , (cid:8) Mg O (cid:9) , { MgO } (cid:1) O + Mg O Mg O k = k + N , M ( MgO , , , T ) O + Mg O Mg O k = k + N , M ( MgO , , , T ) O + Mg O Mg O k = k + N , M ( MgO , , , T ) O + Mg O Mg O k = k + N , M ( MgO , , , T ) O + Mg O Mg O k = k + N , M ( MgO , , , T ) O + Mg O Mg O k = k + N , M ( MgO , , , T ) O + Mg O Mg O k = k + N , M ( MgO , , , T ) MNRAS000
No. Reaction Rate coefficient (cm ( N − ) s − ) with N number of reactants Ref.284 HCO + + e – CO + H k = . · − ( T / ) − . + + Fe Fe + + HCO k = . · − + SiF + + H k = . · − ( T / ) − . S + S k = . · − + + e – S + H k = · − ( T / ) − . CR He + + e – k = . ζ – He + + e – + e – k = exp (cid:16) − . · − ln (cid:16) T e (cid:17) + . (cid:16) T e (cid:17) − . (cid:16) T e (cid:17) + . (cid:16) T e (cid:17) − . (cid:16) T e (cid:17) + . (cid:16) T e (cid:17) − . (cid:16) T e (cid:17) + . (cid:16) T e (cid:17) − . (cid:17) + + e – He + γ k = . · − ( T / ) − . + + HF F + + H + He k = . · − ( T / ) − . + + Si Si + + He k = . · − + + SiO O + Si + + He k = · − + + e – He + H k = · − ( T / ) − . – Mg + + e – + e – k = . · − (cid:16) T (cid:17) . O MgOH + H k = · − EQR (cid:0) T , (cid:8) H O , Mg (cid:9) , { H , MgOH } (cid:1) + HCO + Mg + k = . · − MgO + NO k = . · − exp (cid:16) − . R kJ T (cid:17) MgO + O k = . · − T . EQR (cid:0) T , (cid:8) MgO , O (cid:9) , { MgO , O } (cid:1) MgO + O k = . · − exp (cid:16) − T (cid:17) + S + Mg + k = . · − + Si + Mg + k = . · − + SiO + Mg + k = · − + + e – Mg + γ k = . · − ( T / ) − . O MgO + Mg O k = k + N , M ( MgO , , , T ) EQR (cid:0) T , (cid:8) Mg O (cid:9) , (cid:8) Mg O , MgO (cid:9)(cid:1) O Mg O + Mg O k = k + N , M ( MgO , , , T ) EQR (cid:0) T , (cid:8) Mg O (cid:9) , (cid:8) Mg O , Mg O (cid:9)(cid:1) O Mg O + Mg O k = k + N , M ( MgO , , , T ) EQR (cid:0) T , (cid:8) Mg O (cid:9) , (cid:8) Mg O , Mg O (cid:9)(cid:1) O Mg O + Mg O k = k + N , M ( MgO , , , T ) EQR (cid:0) T , (cid:8) Mg O (cid:9) , (cid:8) Mg O , Mg O (cid:9)(cid:1) O Mg O + Mg O k = k + N , M ( MgO , , , T ) EQR (cid:0) T , (cid:8) Mg O (cid:9) , (cid:8) Mg O (cid:9)(cid:1) O MgO + MgO k = k + N , M ( MgO , , , T ) EQR (cid:0) T , (cid:8) Mg O (cid:9) , { MgO } (cid:1) O + Mg O Mg O k = k + N , M ( MgO , , , T ) O + Mg O Mg O k = k + N , M ( MgO , , , T ) O + Mg O Mg O k = k + N , M ( MgO , , , T ) O + Mg O Mg O k = k + N , M ( MgO , , , T ) O + Mg O Mg O k = k + N , M ( MgO , , , T ) O + Mg O Mg O k = k + N , M ( MgO , , , T ) O + Mg O Mg O k = k + N , M ( MgO , , , T ) MNRAS000 , 1–68 (2019) ucleation in AGB winds No. Reaction Rate coefficient (cm ( N − ) s − ) with N number of reactants Ref.319 Mg O MgO + Mg O k = k + N , M ( MgO , , , T ) EQR (cid:0) T , (cid:8) Mg O (cid:9) , (cid:8) Mg O , MgO (cid:9)(cid:1) O + Mg O Mg O k = k + N , M ( MgO , , , T ) O + Mg O Mg O k = k + N , M ( MgO , , , T ) O + Mg O Mg O k = k + N , M ( MgO , , , T ) O + Mg O Mg O k = k + N , M ( MgO , , , T ) O + Mg O Mg O k = k + N , M ( MgO , , , T ) O MgO + Mg O k = k + N , M ( MgO , , , T ) EQR (cid:0) T , (cid:8) Mg O (cid:9) , (cid:8) Mg O , MgO (cid:9)(cid:1) O Mg O + Mg O k = k + N , M ( MgO , , , T ) EQR (cid:0) T , (cid:8) Mg O (cid:9) , (cid:8) Mg O (cid:9)(cid:1) O + Mg O Mg O k = k + N , M ( MgO , , , T ) O + Mg O Mg O k = k + N , M ( MgO , , , T ) O + Mg O Mg O k = k + N , M ( MgO , , , T ) O MgO + Mg O k = k + N , M ( MgO , , , T ) EQR (cid:0) T , (cid:8) Mg O (cid:9) , (cid:8) Mg O , MgO (cid:9)(cid:1) O Mg O + Mg O k = k + N , M ( MgO , , , T ) EQR (cid:0) T , (cid:8) Mg O (cid:9) , (cid:8) Mg O , Mg O (cid:9)(cid:1) O + Mg O Mg O k = k + N , M ( MgO , , , T ) O MgO + Mg O k = k + N , M ( MgO , , , T ) EQR (cid:0) T , (cid:8) Mg O (cid:9) , (cid:8) Mg O , MgO (cid:9)(cid:1) O Mg O + Mg O k = k + N , M ( MgO , , , T ) EQR (cid:0) T , (cid:8) Mg O (cid:9) , (cid:8) Mg O , Mg O (cid:9)(cid:1) O Mg O + Mg O k = k + N , M ( MgO , , , T ) EQR (cid:0) T , (cid:8) Mg O (cid:9) , (cid:8) Mg O (cid:9)(cid:1) O MgO + Mg O k = k + N , M ( MgO , , , T ) EQR (cid:0) T , (cid:8) Mg O (cid:9) , (cid:8) Mg O , MgO (cid:9)(cid:1) O Mg O + Mg O k = k + N , M ( MgO , , , T ) EQR (cid:0) T , (cid:8) Mg O (cid:9) , (cid:8) Mg O , Mg O (cid:9)(cid:1) O Mg O + Mg O k = k + N , M ( MgO , , , T ) EQR (cid:0) T , (cid:8) Mg O (cid:9) , (cid:8) Mg O , Mg O (cid:9)(cid:1) O MgO + Mg O k = k + N , M ( MgO , , , T ) EQR (cid:0) T , (cid:8) Mg O (cid:9) , (cid:8) Mg O , MgO (cid:9)(cid:1) O Mg O + Mg O k = k + N , M ( MgO , , , T ) EQR (cid:0) T , (cid:8) Mg O (cid:9) , (cid:8) Mg O , Mg O (cid:9)(cid:1) O Mg O + Mg O k = k + N , M ( MgO , , , T ) EQR (cid:0) T , (cid:8) Mg O (cid:9) , (cid:8) Mg O , Mg O (cid:9)(cid:1) O Mg O + Mg O k = k + N , M ( MgO , , , T ) EQR (cid:0) T , (cid:8) Mg O (cid:9) , (cid:8) Mg O (cid:9)(cid:1) O MgO + Mg O k = k + N , M ( MgO , , , T ) EQR (cid:0) T , (cid:8) Mg O (cid:9) , (cid:8) Mg O , MgO (cid:9)(cid:1) O Mg O + Mg O k = k + N , M ( MgO , , , T ) EQR (cid:0) T , (cid:8) Mg O (cid:9) , (cid:8) Mg O , Mg O (cid:9)(cid:1) O Mg O + Mg O k = k + N , M ( MgO , , , T ) EQR (cid:0) T , (cid:8) Mg O (cid:9) , (cid:8) Mg O , Mg O (cid:9)(cid:1) O Mg O + Mg O k = k + N , M ( MgO , , , T ) EQR (cid:0) T , (cid:8) Mg O (cid:9) , (cid:8) Mg O , Mg O (cid:9)(cid:1) + M MgO + CO + M k = k Troe (cid:16) − .
494 log (cid:16) T (cid:17) + .
827 log (cid:16) T (cid:17) − . , . · − exp (cid:16) − T (cid:17) , . , n tot (cid:17) · EQR (cid:16) T , (cid:110) MgCO (cid:111) , (cid:110) CO , MgO (cid:111)(cid:17) + M MgCO + M k = k Troe (cid:16) − .
494 log (cid:16) T (cid:17) + .
827 log (cid:16) T (cid:17) − . , . · − exp (cid:16) − T (cid:17) , . , n tot (cid:17) O + M MgO H + M k = k Troe (cid:16) − .
127 log (cid:16) T (cid:17) + .
894 log (cid:16) T (cid:17) − . , . · − exp (cid:16) − T (cid:17) , . , n tot (cid:17) O Mg O k = k + N , M ( MgO , , , T ) O Mg O k = k + N , M ( MgO , , , T ) O Mg O k = k + N , M ( MgO , , , T ) MNRAS , 1–68 (2019) J. Boulangier et al.
No. Reaction Rate coefficient (cm ( N − ) s − ) with N number of reactants Ref.353 MgO + Mg O Mg O k = k + N , M ( MgO , , , T ) O Mg O k = k + N , M ( MgO , , , T ) O Mg O k = k + N , M ( MgO , , , T ) O Mg O k = k + N , M ( MgO , , , T ) O Mg O k = k + N , M ( MgO , , , T ) O k = k + N , M ( MgO , , , T ) k = . · − exp (cid:16) − . R kJ T (cid:17) EQR (cid:0) T , { MgO , NO } , (cid:8) Mg , NO (cid:9)(cid:1) k = . · − T . Mg + O k = . · − exp (cid:16) − T (cid:17) EQR (cid:0) T , (cid:8) MgO , O (cid:9) , (cid:8) Mg , O (cid:9)(cid:1) + M MgO + M k = k Troe (cid:16) − .
683 log (cid:16) T (cid:17) + .
423 log (cid:16) T (cid:17) − . , . · − exp (cid:16) − T (cid:17) , . , n tot (cid:17) MgO + O k = . · − exp (cid:16) − T (cid:17) + O MgO + O k = . · − T . + O + M MgO + M k = k Troe (cid:16) T − . . · − , . · − exp (cid:16) − T (cid:17) , . , n tot (cid:17) H + M MgO + H O + M k = k Troe (cid:16) − .
127 log (cid:16) T (cid:17) + .
894 log (cid:16) T (cid:17) − . , . · − exp (cid:16) − T (cid:17) , . , n tot (cid:17) · EQR (cid:16) T , (cid:110) MgO H (cid:111) , (cid:110) H O , MgO (cid:111)(cid:17) H + H MgOH + H O k = · − exp (cid:16) − T (cid:17) + H MgOH + O k = · − + H O MgO H + O k = · − + O MgO + O k = · − O k = · − O MgO H + H k = · − exp (cid:16) − T (cid:17) EQR (cid:0) T , (cid:8) H O , MgOH (cid:9) , (cid:8) H , MgO H (cid:9)(cid:1) CN + C k = · − + C k = · − ( T / ) . NO + CO k = . · − exp (cid:16) − T (cid:17) CR N + + e – k = . ζ = . · − ( T / ) . exp (cid:16) − T (cid:17) = · − = . · − ( T / ) . exp (cid:16) − T (cid:17) + H k = . · − + O k = . · − ( T / ) − . exp (cid:16) . T (cid:17) NO + O k = . · − ( T / ) . exp (cid:16) − T (cid:17) = . · − ( T / ) . exp (cid:16) − T (cid:17) = . · − ( T / ) − . exp (cid:16) − . T (cid:17) k = · − = · − ( T / ) − . = . · − MNRAS000
894 log (cid:16) T (cid:17) − . , . · − exp (cid:16) − T (cid:17) , . , n tot (cid:17) · EQR (cid:16) T , (cid:110) MgO H (cid:111) , (cid:110) H O , MgO (cid:111)(cid:17) H + H MgOH + H O k = · − exp (cid:16) − T (cid:17) + H MgOH + O k = · − + H O MgO H + O k = · − + O MgO + O k = · − O k = · − O MgO H + H k = · − exp (cid:16) − T (cid:17) EQR (cid:0) T , (cid:8) H O , MgOH (cid:9) , (cid:8) H , MgO H (cid:9)(cid:1) CN + C k = · − + C k = · − ( T / ) . NO + CO k = . · − exp (cid:16) − T (cid:17) CR N + + e – k = . ζ = . · − ( T / ) . exp (cid:16) − T (cid:17) = · − = . · − ( T / ) . exp (cid:16) − T (cid:17) + H k = . · − + O k = . · − ( T / ) − . exp (cid:16) . T (cid:17) NO + O k = . · − ( T / ) . exp (cid:16) − T (cid:17) = . · − ( T / ) . exp (cid:16) − T (cid:17) = . · − ( T / ) − . exp (cid:16) − . T (cid:17) k = · − = · − ( T / ) − . = . · − MNRAS000 , 1–68 (2019) ucleation in AGB winds No. Reaction Rate coefficient (cm ( N − ) s − ) with N number of reactants Ref.388 N + SO NS + O k = . · − ( T / ) . exp (cid:16) − T (cid:17) = . · − ( T / ) . exp (cid:16) − T (cid:17) + NO + Si + k = . · − + + e – N + γ k = . · − ( T / ) − . exp (cid:16) . T (cid:17) CR N + N k = ζ = . · − = . · − = · − = . · − ( T / ) . exp (cid:16) − T (cid:17) + + e – N + H k = . · − ( T / ) − . + Fe + Na + k = · − + Mg + Na + k = · − + S + Na + k = . · − + Si + Na + k = . · − + + e – Na + γ k = . · − ( T / ) − . CO + C k = · − ( T / ) − . = . · − exp (cid:16) − T (cid:17) = . · − CR O + + e – k = . ζ = . · − ( T / ) . exp (cid:16) − T (cid:17) = . · − ( T / ) − . exp (cid:16) − T (cid:17) – O – + γ k = . · − O OH + OH k = . · − ( T / ) . exp (cid:16) − T (cid:17) = . · − ( T / ) . exp (cid:16) − T (cid:17) = . · − exp (cid:16) − T (cid:17) = . · − ( T / ) . exp (cid:16) − T (cid:17) = . · − ( T / ) . exp (cid:16) − T (cid:17) = . · − ( T / ) − . exp (cid:16) − . T (cid:17) NO + N k = . · − exp (cid:16) − T (cid:17) = · − + γ k = . · − ( T / ) . + O O + O k = . · − T − . + H k = . · − ( T / ) − . exp (cid:16) − . T (cid:17) k = . · − exp (cid:16) − T (cid:17) SO + O k = . · − exp (cid:16) − T (cid:17) γ k = . · − ( T / ) . + O + Si + k = · − + + e – O + γ k = . · − ( T / ) − . + + Fe Fe + + O k = . · − MNRAS , 1–68 (2019) J. Boulangier et al.
No. Reaction Rate coefficient (cm ( N − ) s − ) with N number of reactants Ref.427 O – + Fe + O + Fe k = . · − ( T / ) − . – + H + O + H k = . · − ( T / ) − . – + Mg + O + Mg k = . · − ( T / ) − . + S SO + O k = . · − ( T / ) . exp (cid:16) . T (cid:17) + O O + O k = · − exp (cid:16) − T (cid:17) = · − exp (cid:16) − T (cid:17) = · − + H k = . · − exp (cid:16) − T (cid:17) = · − = . · − = . · − S HS + H O k = . · − exp (cid:16) − T (cid:17) O + O k = . · − ( T / ) . exp (cid:16) − T (cid:17) = . · − + H k = . · − = · − + SiO + + H k = . · − ( T / ) − . + H k = · − + + e – O + H k = . · − ( T / ) − . PO + O k = · − + + e – P + γ k = . · − ( T / ) − . – S – + γ k = · − + H k = . · − SO + SO k = . · − exp (cid:16) − T (cid:17) + + e – S + γ k = . · − ( T / ) − . + + Fe Fe + + S k = . · − = . · − exp (cid:16) − T (cid:17) SiO + CO k = . · − exp (cid:16) − T (cid:17) + SiH + + CO k = . · − = · − ( T / ) − . exp (cid:16) − T (cid:17) SiO + O k = . · − ( T / ) − . exp (cid:16) − T (cid:17) + P + Si + k = · − + S + Si + k = . · − + + e – Si + γ k = . · − ( T / ) − . + + Fe Fe + + Si k = . · − O SiO + Si O k = k + N , M ( SiO , , , T ) EQR (cid:0) T , (cid:8) Si O (cid:9) , (cid:8) Si O , SiO (cid:9)(cid:1) O Si O + Si O k = k + N , M ( SiO , , , T ) EQR (cid:0) T , (cid:8) Si O (cid:9) , (cid:8) Si O , Si O (cid:9)(cid:1) O Si O + Si O k = k + N , M ( SiO , , , T ) EQR (cid:0) T , (cid:8) Si O (cid:9) , (cid:8) Si O , Si O (cid:9)(cid:1) O Si O + Si O k = k + N , M ( SiO , , , T ) EQR (cid:0) T , (cid:8) Si O (cid:9) , (cid:8) Si O , Si O (cid:9)(cid:1) MNRAS000
No. Reaction Rate coefficient (cm ( N − ) s − ) with N number of reactants Ref.427 O – + Fe + O + Fe k = . · − ( T / ) − . – + H + O + H k = . · − ( T / ) − . – + Mg + O + Mg k = . · − ( T / ) − . + S SO + O k = . · − ( T / ) . exp (cid:16) . T (cid:17) + O O + O k = · − exp (cid:16) − T (cid:17) = · − exp (cid:16) − T (cid:17) = · − + H k = . · − exp (cid:16) − T (cid:17) = · − = . · − = . · − S HS + H O k = . · − exp (cid:16) − T (cid:17) O + O k = . · − ( T / ) . exp (cid:16) − T (cid:17) = . · − + H k = . · − = · − + SiO + + H k = . · − ( T / ) − . + H k = · − + + e – O + H k = . · − ( T / ) − . PO + O k = · − + + e – P + γ k = . · − ( T / ) − . – S – + γ k = · − + H k = . · − SO + SO k = . · − exp (cid:16) − T (cid:17) + + e – S + γ k = . · − ( T / ) − . + + Fe Fe + + S k = . · − = . · − exp (cid:16) − T (cid:17) SiO + CO k = . · − exp (cid:16) − T (cid:17) + SiH + + CO k = . · − = · − ( T / ) − . exp (cid:16) − T (cid:17) SiO + O k = . · − ( T / ) − . exp (cid:16) − T (cid:17) + P + Si + k = · − + S + Si + k = . · − + + e – Si + γ k = . · − ( T / ) − . + + Fe Fe + + Si k = . · − O SiO + Si O k = k + N , M ( SiO , , , T ) EQR (cid:0) T , (cid:8) Si O (cid:9) , (cid:8) Si O , SiO (cid:9)(cid:1) O Si O + Si O k = k + N , M ( SiO , , , T ) EQR (cid:0) T , (cid:8) Si O (cid:9) , (cid:8) Si O , Si O (cid:9)(cid:1) O Si O + Si O k = k + N , M ( SiO , , , T ) EQR (cid:0) T , (cid:8) Si O (cid:9) , (cid:8) Si O , Si O (cid:9)(cid:1) O Si O + Si O k = k + N , M ( SiO , , , T ) EQR (cid:0) T , (cid:8) Si O (cid:9) , (cid:8) Si O , Si O (cid:9)(cid:1) MNRAS000 , 1–68 (2019) ucleation in AGB winds No. Reaction Rate coefficient (cm ( N − ) s − ) with N number of reactants Ref.466 Si O Si O + Si O k = k + N , M ( SiO , , , T ) EQR (cid:0) T , (cid:8) Si O (cid:9) , (cid:8) Si O (cid:9)(cid:1) O SiO + SiO k = k + N , M ( SiO , , , T ) EQR (cid:0) T , (cid:8) Si O (cid:9) , { SiO } (cid:1) O + Si O Si O k = k + N , M ( SiO , , , T ) O + Si O Si O k = k + N , M ( SiO , , , T ) O + Si O Si O k = k + N , M ( SiO , , , T ) O + Si O Si O k = k + N , M ( SiO , , , T ) O + Si O Si O k = k + N , M ( SiO , , , T ) O + Si O Si O k = k + N , M ( SiO , , , T ) O + Si O Si O k = k + N , M ( SiO , , , T ) O SiO + Si O k = k + N , M ( SiO , , , T ) EQR (cid:0) T , (cid:8) Si O (cid:9) , (cid:8) Si O , SiO (cid:9)(cid:1) O + Si O Si O k = k + N , M ( SiO , , , T ) O + Si O Si O k = k + N , M ( SiO , , , T ) O + Si O Si O k = k + N , M ( SiO , , , T ) O + Si O Si O k = k + N , M ( SiO , , , T ) O + Si O Si O k = k + N , M ( SiO , , , T ) O SiO + Si O k = k + N , M ( SiO , , , T ) EQR (cid:0) T , (cid:8) Si O (cid:9) , (cid:8) Si O , SiO (cid:9)(cid:1) O Si O + Si O k = k + N , M ( SiO , , , T ) EQR (cid:0) T , (cid:8) Si O (cid:9) , (cid:8) Si O (cid:9)(cid:1) O + Si O Si O k = k + N , M ( SiO , , , T ) O + Si O Si O k = k + N , M ( SiO , , , T ) O + Si O Si O k = k + N , M ( SiO , , , T ) O SiO + Si O k = k + N , M ( SiO , , , T ) EQR (cid:0) T , (cid:8) Si O (cid:9) , (cid:8) Si O , SiO (cid:9)(cid:1) O Si O + Si O k = k + N , M ( SiO , , , T ) EQR (cid:0) T , (cid:8) Si O (cid:9) , (cid:8) Si O , Si O (cid:9)(cid:1) O + Si O Si O k = k + N , M ( SiO , , , T ) O SiO + Si O k = k + N , M ( SiO , , , T ) EQR (cid:0) T , (cid:8) Si O (cid:9) , (cid:8) Si O , SiO (cid:9)(cid:1) O Si O + Si O k = k + N , M ( SiO , , , T ) EQR (cid:0) T , (cid:8) Si O (cid:9) , (cid:8) Si O , Si O (cid:9)(cid:1) O Si O + Si O k = k + N , M ( SiO , , , T ) EQR (cid:0) T , (cid:8) Si O (cid:9) , (cid:8) Si O (cid:9)(cid:1) O SiO + Si O k = k + N , M ( SiO , , , T ) EQR (cid:0) T , (cid:8) Si O (cid:9) , (cid:8) Si O , SiO (cid:9)(cid:1) O Si O + Si O k = k + N , M ( SiO , , , T ) EQR (cid:0) T , (cid:8) Si O (cid:9) , (cid:8) Si O , Si O (cid:9)(cid:1) O Si O + Si O k = k + N , M ( SiO , , , T ) EQR (cid:0) T , (cid:8) Si O (cid:9) , (cid:8) Si O , Si O (cid:9)(cid:1) O SiO + Si O k = k + N , M ( SiO , , , T ) EQR (cid:0) T , (cid:8) Si O (cid:9) , (cid:8) Si O , SiO (cid:9)(cid:1) O Si O + Si O k = k + N , M ( SiO , , , T ) EQR (cid:0) T , (cid:8) Si O (cid:9) , (cid:8) Si O , Si O (cid:9)(cid:1) O Si O + Si O k = k + N , M ( SiO , , , T ) EQR (cid:0) T , (cid:8) Si O (cid:9) , (cid:8) Si O , Si O (cid:9)(cid:1) O Si O + Si O k = k + N , M ( SiO , , , T ) EQR (cid:0) T , (cid:8) Si O (cid:9) , (cid:8) Si O (cid:9)(cid:1) O SiO + Si O k = k + N , M ( SiO , , , T ) EQR (cid:0) T , (cid:8) Si O (cid:9) , (cid:8) Si O , SiO (cid:9)(cid:1) O Si O + Si O k = k + N , M ( SiO , , , T ) EQR (cid:0) T , (cid:8) Si O (cid:9) , (cid:8) Si O , Si O (cid:9)(cid:1) O Si O + Si O k = k + N , M ( SiO , , , T ) EQR (cid:0) T , (cid:8) Si O (cid:9) , (cid:8) Si O , Si O (cid:9)(cid:1) O Si O + Si O k = k + N , M ( SiO , , , T ) EQR (cid:0) T , (cid:8) Si O (cid:9) , (cid:8) Si O , Si O (cid:9)(cid:1) + + e – Si + F k = · − ( T / ) − . + + e – Si + H k = · − ( T / ) − . O Si O k = k + N , M ( SiO , , , T ) MNRAS , 1–68 (2019) J. Boulangier et al.
No. Reaction Rate coefficient (cm ( N − ) s − ) with N number of reactants Ref.506 SiO + Si O Si O k = k + N , M ( SiO , , , T ) O Si O k = k + N , M ( SiO , , , T ) O Si O k = k + N , M ( SiO , , , T ) O Si O k = k + N , M ( SiO , , , T ) O Si O k = k + N , M ( SiO , , , T ) O Si O k = k + N , M ( SiO , , , T ) O Si O k = k + N , M ( SiO , , , T ) O k = k + N , M ( SiO , , , T ) + + e – Si + O k = · − ( T / ) − . + + Fe Fe + + SiO k = · − + H SiO + OH k = · − EQR (cid:0) T , (cid:8) SiO , H (cid:9) , { SiO , OH } (cid:1) TiO + CO k = · − exp (cid:16) − . R kJ T (cid:17) O TiO + N k = . · − exp (cid:16) − . R kJ T (cid:17) = . · − exp (cid:16) − . R kJ T (cid:17) TiO + NO k = · − TiO + O k = . · − exp (cid:16) − . R kJ T (cid:17) TiO + SO k = . · − exp (cid:16) − . R kJ T (cid:17) O TiO + Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) EQR (cid:0) T , (cid:8) Ti O (cid:9) , (cid:8) Ti O , TiO (cid:9)(cid:1) O Ti O + Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) EQR (cid:0) T , (cid:8) Ti O (cid:9) , (cid:8) Ti O , Ti O (cid:9)(cid:1) O Ti O + Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) EQR (cid:0) T , (cid:8) Ti O (cid:9) , (cid:8) Ti O , Ti O (cid:9)(cid:1) O Ti O + Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) EQR (cid:0) T , (cid:8) Ti O (cid:9) , (cid:8) Ti O , Ti O (cid:9)(cid:1) O Ti O + Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) EQR (cid:0) T , (cid:8) Ti O (cid:9) , (cid:8) Ti O (cid:9)(cid:1) O TiO + TiO k = k + N , M (cid:0) TiO , , , T (cid:1) EQR (cid:0) T , (cid:8) Ti O (cid:9) , (cid:8) TiO (cid:9)(cid:1) O + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) O + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) O + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) O + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) O + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) O + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) O + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) O TiO + Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) EQR (cid:0) T , (cid:8) Ti O (cid:9) , (cid:8) Ti O , TiO (cid:9)(cid:1) O + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) O + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) O + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) O + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) O + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) O TiO + Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) EQR (cid:0) T , (cid:8) Ti O (cid:9) , (cid:8) Ti O , TiO (cid:9)(cid:1) O Ti O + Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) EQR (cid:0) T , (cid:8) Ti O (cid:9) , (cid:8) Ti O (cid:9)(cid:1) O + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) O + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) MNRAS000
No. Reaction Rate coefficient (cm ( N − ) s − ) with N number of reactants Ref.506 SiO + Si O Si O k = k + N , M ( SiO , , , T ) O Si O k = k + N , M ( SiO , , , T ) O Si O k = k + N , M ( SiO , , , T ) O Si O k = k + N , M ( SiO , , , T ) O Si O k = k + N , M ( SiO , , , T ) O Si O k = k + N , M ( SiO , , , T ) O Si O k = k + N , M ( SiO , , , T ) O k = k + N , M ( SiO , , , T ) + + e – Si + O k = · − ( T / ) − . + + Fe Fe + + SiO k = · − + H SiO + OH k = · − EQR (cid:0) T , (cid:8) SiO , H (cid:9) , { SiO , OH } (cid:1) TiO + CO k = · − exp (cid:16) − . R kJ T (cid:17) O TiO + N k = . · − exp (cid:16) − . R kJ T (cid:17) = . · − exp (cid:16) − . R kJ T (cid:17) TiO + NO k = · − TiO + O k = . · − exp (cid:16) − . R kJ T (cid:17) TiO + SO k = . · − exp (cid:16) − . R kJ T (cid:17) O TiO + Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) EQR (cid:0) T , (cid:8) Ti O (cid:9) , (cid:8) Ti O , TiO (cid:9)(cid:1) O Ti O + Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) EQR (cid:0) T , (cid:8) Ti O (cid:9) , (cid:8) Ti O , Ti O (cid:9)(cid:1) O Ti O + Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) EQR (cid:0) T , (cid:8) Ti O (cid:9) , (cid:8) Ti O , Ti O (cid:9)(cid:1) O Ti O + Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) EQR (cid:0) T , (cid:8) Ti O (cid:9) , (cid:8) Ti O , Ti O (cid:9)(cid:1) O Ti O + Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) EQR (cid:0) T , (cid:8) Ti O (cid:9) , (cid:8) Ti O (cid:9)(cid:1) O TiO + TiO k = k + N , M (cid:0) TiO , , , T (cid:1) EQR (cid:0) T , (cid:8) Ti O (cid:9) , (cid:8) TiO (cid:9)(cid:1) O + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) O + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) O + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) O + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) O + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) O + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) O + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) O TiO + Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) EQR (cid:0) T , (cid:8) Ti O (cid:9) , (cid:8) Ti O , TiO (cid:9)(cid:1) O + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) O + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) O + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) O + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) O + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) O TiO + Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) EQR (cid:0) T , (cid:8) Ti O (cid:9) , (cid:8) Ti O , TiO (cid:9)(cid:1) O Ti O + Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) EQR (cid:0) T , (cid:8) Ti O (cid:9) , (cid:8) Ti O (cid:9)(cid:1) O + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) O + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) MNRAS000 , 1–68 (2019) ucleation in AGB winds No. Reaction Rate coefficient (cm ( N − ) s − ) with N number of reactants Ref.546 Ti O + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) O TiO + Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) EQR (cid:0) T , (cid:8) Ti O (cid:9) , (cid:8) Ti O , TiO (cid:9)(cid:1) O Ti O + Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) EQR (cid:0) T , (cid:8) Ti O (cid:9) , (cid:8) Ti O , Ti O (cid:9)(cid:1) O + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) O TiO + Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) EQR (cid:0) T , (cid:8) Ti O (cid:9) , (cid:8) Ti O , TiO (cid:9)(cid:1) O Ti O + Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) EQR (cid:0) T , (cid:8) Ti O (cid:9) , (cid:8) Ti O , Ti O (cid:9)(cid:1) O Ti O + Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) EQR (cid:0) T , (cid:8) Ti O (cid:9) , (cid:8) Ti O (cid:9)(cid:1) O TiO + Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) EQR (cid:0) T , (cid:8) Ti O (cid:9) , (cid:8) Ti O , TiO (cid:9)(cid:1) O Ti O + Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) EQR (cid:0) T , (cid:8) Ti O (cid:9) , (cid:8) Ti O , Ti O (cid:9)(cid:1) O Ti O + Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) EQR (cid:0) T , (cid:8) Ti O (cid:9) , (cid:8) Ti O , Ti O (cid:9)(cid:1) O TiO + Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) EQR (cid:0) T , (cid:8) Ti O (cid:9) , (cid:8) Ti O , TiO (cid:9)(cid:1) O Ti O + Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) EQR (cid:0) T , (cid:8) Ti O (cid:9) , (cid:8) Ti O , Ti O (cid:9)(cid:1) O Ti O + Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) EQR (cid:0) T , (cid:8) Ti O (cid:9) , (cid:8) Ti O , Ti O (cid:9)(cid:1) O Ti O + Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) EQR (cid:0) T , (cid:8) Ti O (cid:9) , (cid:8) Ti O (cid:9)(cid:1) O TiO + Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) EQR (cid:0) T , (cid:8) Ti O (cid:9) , (cid:8) Ti O , TiO (cid:9)(cid:1) O Ti O + Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) EQR (cid:0) T , (cid:8) Ti O (cid:9) , (cid:8) Ti O , Ti O (cid:9)(cid:1) O Ti O + Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) EQR (cid:0) T , (cid:8) Ti O (cid:9) , (cid:8) Ti O , Ti O (cid:9)(cid:1) O Ti O + Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) EQR (cid:0) T , (cid:8) Ti O (cid:9) , (cid:8) Ti O , Ti O (cid:9)(cid:1) k = · − exp (cid:16) − . R kJ T (cid:17) EQR (cid:0) T , { CO , TiO } , (cid:8) CO , Ti (cid:9)(cid:1) = . · − exp (cid:16) − . R kJ T (cid:17) EQR ( T , { N , TiO } , { NO , Ti }) Ti + N O k = . · − exp (cid:16) − . R kJ T (cid:17) EQR (cid:0) T , (cid:8) N , TiO (cid:9) , (cid:8) N O , Ti (cid:9)(cid:1) k = · − EQR (cid:0) T , { NO , TiO } , (cid:8) NO , Ti (cid:9)(cid:1) + N k = . · − k = . · − exp (cid:16) − . R kJ T (cid:17) EQR (cid:0) T , { O , TiO } , (cid:8) O , Ti (cid:9)(cid:1) TiO + O k = . · − + H k = . · − ( T ) . k = . · − exp (cid:16) − . R kJ T (cid:17) EQR (cid:0) T , { SO , TiO } , (cid:8) SO , Ti (cid:9)(cid:1) + H TiO + OH k = · − exp (cid:16) − T (cid:17) + N TiO + NO k = . · − EQR (cid:0) T , (cid:8) N , TiO (cid:9) , { NO , TiO } (cid:1) + O TiO + O k = . · − EQR (cid:0) T , (cid:8) O , TiO (cid:9) , (cid:8) O , TiO (cid:9)(cid:1) + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) + Ti O Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) + TiO Ti O k = k + N , M (cid:0) TiO , , , T (cid:1) MNRAS , 1–68 (2019) J. Boulangier et al.
Parameters: T e = T / . eV K − is the gas temperature in electron volt ζ = . · − s − is the cosmic ray (CR) flux n tot is the total number density of the gas R kJ is the universal gas constant in kJ K − mol − k + N , M ( X , N , M , T ) = π ( N / r X + M / r X ) (cid:113) k B T πµ N , M , µ N , M = m XN m XM m XN + m XM EQR ( T , R , P) = (cid:16) P ◦ k B T (cid:17) ∆ s exp (cid:16) (cid:205) r ∈R G ◦ r − (cid:205) p ∈P G ◦ p R kJ T (cid:17) , ∆ s = |P| − |R| , P ◦ = Pak
Troe ( k , k ∞ , F c , n tot ) = k n tot + k n tot k ∞ F β c , β = + (cid:16) log k n tot k ∞ (cid:17) References: (1) Washburn et al. (2008), (2) Sharipov et al. (2012), (3) Starik et al. (2014), (4) Sharipov et al. (2011),(5) Sharipov & Starik (2016), (6) This work, (7) Swihart et al. (2003), (8)
UMIST database McElroy et al. (2013) , (9)
KIDA database Wakelam et al. (2012), (10) Verner & Ferland (1996), (11) Janev et al. (1987), (12) Forrey (2013), (13) Glover& Abel (2008), (14) DeMore et al. (1997), (15) Abel et al. (1997), (16) Poulaert et al. (1978), (17) Capitelli et al. (2007),(18) Glover et al. (2010), (19) Verner & Ferland (1996), (20) Plane et al. (2015), (21) Whalley et al. (2011), (22) Plane &Whalley (2012), (23) Plane & Helmer (1995), (24) Rollason & Plane (2001), (25) Langowski et al. (2015), (26) Atkinsonet al. (2004), (27) Campbell & McClean (1993), (28) Ritter & Weisshaar (1989), (29) Higuchi et al. (2008) and (30) Plane (2013)
Notes:
The reactants M act as catalyists and can be any species. Therefore the total number density of the gas n tot is used as its density. References of reactions that contain the equilibrium ratio function EQR refer to the references of thereversed reaction. Not all reaction rate coefficients are valid in the considered temperature range. However, due to a lack ofliterature data, those coeffiecients are extrapolated in temperature when necessary.
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