THE CHEMISTRY OF POPULATION III SUPERNOVA EJECTA. II. THE NUCLEATION OF MOLECULAR CLUSTERS AS A DIAGNOSTIC FOR DUST IN THE EARLY UNIVERSE

In space, the 20 μm band of SiO₂ was detected in Herbig Ae/Be stars with the Infrared Space Observatory (ISO; Bouwman et al. 2001), while the modeling of Cas A IR spectra from Spitzer suggests SiO₂ as a potential dust candidate (Rho et al. 2008). In the laboratory, silica clusters form following two routes depending on the hydrogen content of the gas from which they nucleate. Where H is present, Zachariah & Burguess (1994) report the importance of the OH hydroxyl radical in the formation of silica clusters in H/O/Ar-rich flames fueled with silane (SiH₄), using laser-induced fluorescence techniques to characterize the gas phase species entering in the conversion of gas to particle processes. Indeed, OH triggers the formation of gaseous SiO₂ through the reactions (see CD09 for more detail)

$$\begin{equation} \rm SiO + OH \longrightarrow SiO_2 + H, \end{equation} \tag{ 1 }$$

$$\begin{equation} \rm SiO + H_2O \longrightarrow SiO_2 + H_2, \end{equation} \tag{ 2 }$$

and consequent cluster growth occurs from the SiO₂ polymerization reaction

$$\begin{equation} {\rm SiO}_2 + ({\rm SiO}_2)_n \longrightarrow ({\rm SiO}_{2})_{n+1}, \end{equation} \tag{ 3 }$$

where 1 ⩽ n ⩽ 4.

When the gas is hydrogen poor, the nucleation of silica occurs through direct polymerization of SiO to form small (SiO)_n molecular clusters, precursors to solid silicon monoxide SiO. At temperatures ⩾ 900 K, solid SiO is unstable and directly decomposes to silica SiO₂ and solid Si. We thus consider that half of the total number of (SiO)_m particles will be in the form of (SiO₂)_m, whereas the other half will be in solid (Si)_m. It is not clear whether this disproportionation effect gives rise to two separated condensate populations of SiO₂ and pure Si or to solids made of pure Si clusters embedded in an amorphous silica matrix (Mamiya et al. 2001). We neglect this aspect in the present study and consider that solid SiO gives rise to the formation of two types of solids, whatever their final form. SiO is first synthesized from the Si oxidation reaction (see CD09 for more detail)

$$\begin{equation} \rm Si + O_2 \longrightarrow SiO + O, \end{equation} \tag{ 4 }$$

by reaction with CO

$$\begin{equation} \rm Si + CO \longrightarrow SiO + C, \end{equation} \tag{ 5 }$$

and the radiative association reaction

$$\begin{equation} {\rm Si} + {\rm O} \longrightarrow {\rm SiO} + h\nu, \end{equation} \tag{ 6 }$$

followed by the polymerization of SiO

$$\begin{equation} {\rm SiO} + ({\rm SiO})_n \longrightarrow ({\rm SiO})_{n+1}. \end{equation} \tag{ 7 }$$

with 1 ⩽ n ⩽ 4. In pyrolysis experiments, reaction (7) usually happens at almost collision rate and is at or near high-pressure limit at room temperature. However, as the temperature increases, the rate decreases due to the onset of energy transfer effects. The rate is also highly pressure dependent and is estimated by Zachariah & Tsang (1993, hereafter ZT93) for the high temperatures encountered in the combustion flames and applicable to SN ejecta. For our ejecta, we scale the rates derived by ZT93 for 1 atm pressure according to the total number of collisions per cm⁻³ for the ejecta pressure under consideration. The resulting rates for the polymerization reactions of type reaction (7) are a factor of ∼100 smaller than those of ZT93 at 1 atm pressure. We model both the formation of (SiO₂)_n and (SiO)_m up to n = 4 and m = 5. The most stable form of (SiO₂)₄ has a rhombus chain structure with double oxygen bridged whereas the most stable structure of (SiO)₄ is a buckled eight-membered ring and that of (SiO)₅ has a double ring structure (Lu et al. 2003).

The destruction process for those silica clusters is the thermal fragmentation reaction

$$\begin{equation} ({\rm SiO})_{n} + {\rm M} \longrightarrow ({\rm SiO})_{m} + ({\rm SiO})_{n-m} + {\rm M}, \end{equation} \tag{ 8 }$$

where M represents the ambient gas and 1 ⩽ m ⩽ 3. Theoretical estimation of dissociation energies for silica clusters by Lu et al. (2003) shows that the most common fragmentation products are gas-phase SiO, (SiO)₂, and (SiO)₃, as found in experiments on silica cluster material. This result implies that those species must be rather stable in the gas phase, and for the sake of simplicity, we assume a rate for reaction (8) equals the rate for the thermal fragmentation of SiO (see Table 9 in CD09).

Iron oxide has been proposed as a potential contributor to the 23 μm bands in young stars (Bouwman et al. 2000), in the binary AFGL 4106 system (Molster et al. 1999), and in Cas A (Rho et al. 2008). The reaction of iron with oxygen to generate iron oxides is at the root of fundamental processes in environmental corrosion, biological oxygen transport, or oxide film formation. Of special interest to the present study is the gas phase iron monoxide formation reaction

$$\begin{equation} \rm Fe + O_2 \longrightarrow FeO + O, \end{equation} \tag{ 9 }$$

whose rate was measured at high temperatures by Fontjin et al. (1973) and Akhmadov et al. (1988). The following reaction with carbon dioxide also occurs

$$\begin{equation} \rm Fe + CO_2 \longrightarrow FeO + CO, \end{equation} \tag{ 10 }$$

and its rate was measured by Smirnov (2008). Finally, we consider destruction of iron monoxide through thermal fragmentation and its reaction with atomic oxygen

$$\begin{equation} \rm FeO + O \longrightarrow Fe + O_2, \end{equation} \tag{ 11 }$$

for which a rate was derived by Self & Plane (2003) in their study of iron oxide formation in the Earth upper mesosphere. The thermodynamically stable neutral clusters resulting from laser ablation of an iron rod bathed in an oxidizing gas are of the form (FeO)_m for m ⩽ 10 (Shin et al. 2004). We thus define the nucleation mechanism to small (FeO)_m clusters in the gas phase by successive addition of the FeO monomer

$$\begin{equation} {\rm FeO} + ({\rm FeO})_n \longrightarrow ({\rm FeO})_{n+1}, \end{equation} \tag{ 12 }$$

with 1 ⩽ n ⩽ 3. The rate for this reaction is chosen similar to that of reaction (7). Destruction of small iron oxide clusters proceeds through thermal fragmentation such as reaction (8). Iron oxide clusters are characterized by the following ground-state structures: linear for FeO, rhombus for (FeO)₂, triangular for (FeO)₃, and tetrahedral for (FeO)₄ (Wang et al. 1996; Reilly et al. 2007).

Magnesium oxide, MgO (periclase), was proposed as a component of the dust around the low-redshift quasar PG 2112+059 for its contribution to the 20 μm spectrum region (Markwick-Kemper et al. 2007) and was suggested as potential seeds to silicate condensation in O-rich AGBs (Bhatt & Ford 2007). The nucleation of magnesium oxides involves the formation of MgO molecules in the first place through reaction

$$\begin{equation} \rm Mg + O_2 \longrightarrow MgO + O, \end{equation} \tag{ 13 }$$

and

$$\begin{equation} \rm Mg + CO_2 \longrightarrow MgO + CO, \end{equation} \tag{ 14 }$$

and MgO destruction by thermal fragmentation and reaction with atomic oxygen

$$\begin{equation} \rm MgO + O \longrightarrow Mg + O_2. \end{equation} \tag{ 15 }$$

The rate for reaction (13) was estimated by Kashireninov et al. (1982) whereas those of reactions (14) and (15) were assumed equal to the equivalent reactions involving atomic Fe. We follow the formation of small magnesium oxide clusters (MgO)_n with n = 1–4. As seen from Figure 1, the (MgO)₂ and (MgO)₃ clusters have rhombus and triangular structures, respectively, whereas the lowest energy structure of (MgO)₄ is cubic (Khöler et al. 1997; Bhatt & Ford 2007). We therefore describe MgO cluster formation by the direct polymerization of MgO according to reaction

$$\begin{equation} {\rm MgO} + ({\rm MgO})_n \longrightarrow ({\rm MgO})_{n+1}, \end{equation} \tag{ 16 }$$

where a rate similar to that of reaction (7) has been assumed. Small periclase clusters are destroyed according to the thermal fragmentation reactions similar to reaction (8).

Alumina Al₂O₃ has been detected through its 11 μm band in evolved, oxygen-rich AGB stars (Stencel et al. 1990; Cami 2002) and was found in pre-solar meteorites with an isotopic composition pointing to its O-rich AGB stellar origin (Nittler et al. 1997). Models of the mid-IR excess of the low-redshift quasar PG 2112+059 (Markwick-Kemper et al. 2007), the Blue Luminous Variable Eta Carinae (Chesneau et al. 2005), and the SNR Cas A (Rho et al. 2008) also use alumina as a potential dust component. In the laboratory, laser-induced aluminum evaporation in a helium bath seeded with oxygen produces small aluminum oxide clusters (Desai et al. 1996). Small cluster molecules like Al₂O_x (x = 3–5) show a rhombus structure, including Al₂O₂ rings. The chemical pathway to those small clusters involves the formation of the molecule AlO, and its further reaction with oxygen and itself (Catoire et al. 2003). In our SN ejecta, we only consider the formation of AlO as aluminum has an initial low mass abundance compared to other metals reacting with oxygen (see Tables 2 and 3 of CD09). Furthermore, the formation routes to alumina are still very speculative and we prefer deriving AlO mass yields as a first indication of the alumina content of the ejecta. AlO synthesis occurs through the following reactions:

$$\begin{equation} \rm Al + O_2 \longrightarrow AlO + O, \end{equation} \tag{ 17 }$$

$$\begin{equation} \rm Al+ O + M \longrightarrow AlO + M, \end{equation} \tag{ 18 }$$

$$\begin{equation} \rm Al + CO_2 \longrightarrow AlO + CO, \end{equation} \tag{ 19 }$$

and finally, in the presence of hydrogen,

$$\begin{equation} \rm Al+ H_2O \longrightarrow AlO + OH. \end{equation} \tag{ 20 }$$

Reaction rates for reactions (17)–(20) were derived by Garland & Nelson (1992), Catoire et al. (2003), and McClean et al. (1993). Destruction processes for gas-phase AlO molecules are thermal fragmentation by the ambient gas, collisions with Compton electron, and reaction with He⁺.

The main providers of AC grains in galaxies are carbon-rich AGB stars in the late stages of their evolution, but other formation loci include RCrB stars, the colliding-wind region of carbon-rich Wolf–Rayet binary systems, and SN ejecta. Whenever the carbon content of a gaseous medium is larger than that of oxygen, AC dust may condense. The nucleation to solid clusters involves different carbon routes dependent of the temperature and the hydrogen content of the gas (Cherchneff 2009). When hydrogen is present and for the low temperatures encountered in premixed acetylene flames (T_gas ∼ 1500 K), unsaturated hydrocarbons form, among them acetylene C₂H₂ and its isomeric radical vinylidene (HHCC), the ethynyl radical C₂H, the propargyl radical C₃H₃, and vinylacetylene C₄H₄. The formation of the planar, aromatic ring benzene C₆H₆ results from successive additions of acetylene and its isomer, or the direct reaction of two propargyl radicals (Miller & Melius 1992). Its further growth to PAHs occurs through successive addition of acetylene on aromatic radical sites (Frenklach & Feigelson 1989; Cherchneff et al. 1992; Richter & Howard 2002).

When the environment is hot (T_gas ∼ 4000 K) and hydrogen-free, the nucleation of carbon clusters involves the synthesis of pure carbon chains of sp hybridization up to C₉. For C₁₀, the most stable structure is ringed and the closure of the chain to large monocyclic rings occurs. The coalescence of such rings in bicyclic and tricyclic rings as precursors to fullerene formation is observed in pulsed laser vaporization experiments (von Helden et al. 1993; Hunter et al. 1993) and the condensate is in the form of fullerene-like soot (Jäger et al. 2009).

For the purpose of this study, we explore the kinetics of the formation of carbon chains up to C₉ and the closure to the monocyclic ring C₁₀. As our ejecta are considered hydrogen-free as hydrogen is not microscopically mixed with other constituents, we disregard the formation routes to aromatics involving hydrocarbons. Our kinetic network for chain formation includes the radiative association reactions of the type (Clayton et al. 1999, 2001; Deneault et al. 2006)

$$\begin{equation} {\rm C} + {\rm C}_n \longrightarrow {\rm C}_{n+1} + h\nu, \end{equation} \tag{ 30 }$$

with n = 1–9. For n = 1, Andreazza & Singh (1997) derived a low associative rate (k₃₀ = 1 × 10⁻¹⁷ cm³ s⁻¹) whereas the rate for larger chain synthesis is assumed much faster (k₃₀ = 1 × 10⁻¹⁰ cm³ s⁻¹) as larger chains are able to quickly stabilize their formation complex owing to their numerous vibrational degrees of freedom (Clayton et al. 1999). An additional growth process for chains is their mutual collision and binding by edge carbon atoms. According to Schweigert et al. (1995), such mechanisms have energy barriers less than 0.5 eV. The thermodynamics of the addition reactions are estimated using thermodynamic data for small carbon chains calculated by Clayton et al. (2001) and we find that these additions are all exothermic processes. Therefore, we include the additional growth pathways to chain synthesis

$$\begin{equation} {\rm C}_{n-m} + {\rm C}_m \longrightarrow {\rm C}_{n}, \end{equation} \tag{ 31 }$$

with m = 2–7. The rate for reaction (31) is assumed to be collisional with an estimated value of 1 × 10⁻¹⁰ cm³ s⁻¹.

The dominant destruction channels of carbon clusters included in the model are first, the thermal fragmentation of the chains through collision with the ambient gas

$$\begin{equation} {\rm C}_n + {\rm M} \longrightarrow {\rm C}_{n-m} + {\rm C}_m + {\rm M}, \end{equation} \tag{ 32 }$$

second, the oxidation of the chains through reaction of atomic oxygen with end-cap carbons to form CO

$$\begin{equation} {\rm O} + {\rm C}_n \longrightarrow {\rm C}_{n-1} + {\rm CO}, \end{equation} \tag{ 33 }$$

third, the collision with Compton electrons

$$\begin{equation} {\rm C}_n + e^-_{\rm Compton} \longrightarrow {\rm C}_{n-m} + {\rm C}_m + e^-_{\rm Compton}, \end{equation} \tag{ 34 }$$

and fourth, the destruction by He⁺

$$\begin{equation} {\rm C}_n + {\rm He}^+ \longrightarrow {\rm C}_{n-1} + {\rm C}^+ + {\rm He}. \end{equation} \tag{ 35 }$$

The rates for reactions (32) are those used for C₂ in CD09, while the rate for reactions (33) is that of Clayton et al. (1999, 2001), and finally, the rate for reaction (34) is derived following the formalism explained in CD09.

Silicon carbide grains are ubiquitous in various evolved circumstellar environments (Molster & Waters 2003). In AGB stars, the SiC 11.3 μm band is observed either in emission or absorption for low and high mass loss rates, respectively (Speck et al. 1997), but there is no detection of this band in SN ejecta to date. Presolar SiC grains bearing the isotopic signatures pertaining to SNe are however identified in meteorites (Bernatowicz et al. 1987). In SN ejecta, the formation of SiC grains implies the mixing of Si- and C-rich non-adjacent layers, a situation not found in the dust formation layers of carbon stars where SiO and C-bearing chemical species are simultaneously present in large quantities (Willacy & Cherchneff 1998). In the laboratory, SiC whiskers and nanorods are produced by different methods from vaporization of SiC bulk material, to the surface reaction of gas-phase SiO on carbon films or nanoclusters, or the gas phase formation from chlorosilanes, carbon monoxide, and methane (Zhou & Seraphin 1994). In the present study, we consider the formation of SiC and (SiC)₂ according to the following reactions:

$$\begin{equation} \rm Si + CO \longrightarrow SiC +O, \end{equation} \tag{ 36 }$$

$$\begin{equation} {\rm Si} + {\rm C}_2 \longrightarrow {\rm SiC} +{\rm C}, \end{equation} \tag{ 37 }$$

$$\begin{equation} \rm C + SiO \longrightarrow SiC +O, \end{equation} \tag{ 38 }$$

and

$$\begin{equation} {\rm C} + {\rm Si} \longrightarrow {\rm SiC} + h\nu, \end{equation} \tag{ 39 }$$

followed by the nucleation reaction

$$\begin{equation} \rm SiC + SiC \longrightarrow (SiC)_2. \end{equation} \tag{ 40 }$$

The rate for reaction (40) is assumed to proceed with the rate of reaction (7). The (SiC)₂ cluster has a favored rhombic structure (Yadav et al. 2006) similar to that of (SiO)₂ in Figure 1, but a linear structure is also possible. The destruction channels for SiC are reactions with He⁺, the reverse processes of reactions (36)–(38), and thermal fragmentation, while the latter is the only destruction channel considered for (SiC)₂.

A direct comparison between results for our fully mixed ejecta presented in Section 4.1 and those of NK03 is possible as NK03 ejecta models were used in CD09 and in the present study. NK03 results for the fully mixed ejecta of the 170 M_☉ and 20 M_☉ progenitors are listed in Table 5 and compared to the present results. Despite identical initial conditions for the ejecta chemical composition and similar ejecta gas parameters, the present upper limits for the dust yields are lower than values derived by NK03, with a greater discrepancy (a factor of 2–5) for the 20 M_☉ progenitor. We also see that the chemical composition of the solids formed is totally different. The present models predict the formation of silica and pure silicon dust as the prominent condensates, along with alumina and pure magnesium and iron grains to a less extent. When full depletion of Fe and Mg is considered, fayalite and forsterite also form. On the other hand, NK03 predict the formation of silica, forsterite/enstatite (MgSiO₃), magnetite (Fe₃O₄), and alumina. In their model, iron is trapped in magnetite whereas it is in the form of pure iron clusters in the present model. Also, NK03 do not form pure magnesium grains or pure silicon grains resulting from the disproportionation of silicon monoxide clusters at high temperatures. The only agreement we find regarding the dust composition is related to the lack of carbon and silicon carbide dust in the ejecta. NK03 assume that the whole atomic C is quickly transformed into CO, and we confirm from chemical kinetics that this is indeed the case for the fully microscopically mixed ejecta, whatever the initial mass of the progenitor.

More differences arise in the condensation sequence derived by both studies. NK03 find for both cases that the condensation of alumina proceeds first, followed by forsterite/enstatite, silica, and magnetite. For their 170 M_☉ progenitor, Al₂O₃ condenses at ∼500 days, followed by Mg₂SiO₄ and MgSiO₃ at ∼520 days, silica at ∼550 days, and finally Fe₃O₄ at ∼550 days. These various post-explosion times correspond to epochs when the nucleation rates of the various solids are maximum. However, as mentioned before, these nucleation rates are expressed as a function of physical quantities inappropriate to the description of molecular-size clusters, and the relevance of those condensation sequences has to be seriously questioned. Inspection of Figures 2 and 3 shows different formation times for our cluster precursors, driven by chemical kinetics. In particular, the high initial silicon yield characterizing the 170 M_☉ progenitor fosters a rapid formation of silicon monoxide clusters (hence silica and pure Si precursors) at high temperatures as soon as 400 days, comparable to what is observed in ceramic-forming flames (Zachariah & Burguess 1994). The pure metallic clusters (Mg)₄ and (Fe)₄ on the other hand are forming at t > 700 days for the reasons explained in Section 4.1. Finally, AlO forms at day 900. Those strong discrepancies in formation times reflect the fact that condensation from the gas phase is controlled by non-equilibrium chemically controlled nucleation processes, each of which has its own temperature dependence.

For their 20 M_☉ progenitor, NK03 find that Al₂O₃ condenses first at ∼380 days, followed by Mg₂SiO₄ and MgSiO₃ at ∼400 days, silica at ∼420 days, and finally Fe₃O₄ at ∼430 days. This condensation sequence is similar to that for their PISN case though solid formation proceeds at slightly earlier times owing to the lower ejecta temperatures. In the present study, nucleation times are totally different for the PISN and the CCSN cases, reflecting different chemistries at play in the ejecta. As already stressed in Section 4.1, the large He⁺ content of the 20 M_☉ ejecta totally hampers cluster formation before day 700. Once again, these strong deviations are due to the inappropriate use of CNT applied to kinetically commanded environments.

We run our chemical network including the formation of molecules and nucleation of clusters for the initial elemental yields of the 20 M_☉ progenitor given by Table 2 of TF01 and their specific profiles for the temperature and number density of this ejecta. The elemental composition used by TF01 is that derived from explosion models by Woosley & Weaver (1995) for their zero metallicity (Z = 0) progenitors. From TF01 Table 2, we note for their 20 M_☉ (and 18 M_☉) progenitor: (1) anomalously very low yields for heavy elements such as Si, Mg, S, and Al; and (2) a carbon-to-oxygen ratio (C/O) larger than 1 compared to values for slightly lower or larger progenitor masses. Such very low mass yields for refractory elements and a C/O > 1 are not present in other explosion models like those of UN02 or the more recent calculations of Heger & Woosley (2008) for a similar progenitor mass. Moreover, in those last two studies and for the specific 20 M_☉ progenitor mass, the elemental yields are found to be quite similar. The paucity of refractory elements like Si, Mg, or Al, being all key players in the dust condensation processes, drastically changes the chemical nature of the dust that condenses. We also note that the ejecta temperature profile used by TF01 is derived from comparison with the ejecta of SN1987A. The photospheric temperature is chosen equal to 5400 K and slowly decreases with time to reach 960 K at day 1000, as illustrated in Table 2. The ejecta temperature from 100 to 1000 days post-explosion derived by NK03 and used in the present study and in CD09 takes into account the deposition of energy by radioactive elements through the degradation of γ- and X-rays using a radiative transfer formalism coupled to the energy equation. As seen from Table 2, the resulting temperature variation with post-explosion time has a much sharper decrease with a higher initial temperature at day 100. As to gas number density, TF01 mentioned a value of n = 1 × 10⁸ cm⁻³ for temperatures less than 6000 K but do not provide a specific number density profile for their 20 M_☉ progenitor. We thus assume that the gas number density has the constant value of 10⁸ cm⁻³ over the timespan involved in our test study.

The initial He/C-rich, Si/Mg-poor ejecta elemental composition and temperature profile used by TF01 imply that only carbon dust is expected to form if it can form. Indeed, they find that most of the dust formed for this specific 20 M_☉ progenitor is AC with an AC dust mass yield of ∼ 0.09 M_☉. When using a similar gas elemental composition, temperature, and density profiles for our 20 M_☉ fully mixed ejecta, our results show that the only molecules that are synthesized in the ejecta are C₂ and CO, as illustrated in Figure 4. Their abundances with respect to the total gas number density are very low and comprised between 1 × 10⁻¹³ and 1 × 10⁻¹¹ from day 100 to day 1000 due to the overwhelming presence of He⁺. Indeed, the relatively high ejecta temperatures at day 300 and later foster the ionization of He and impede He⁺ recombination, hence sustaining a strong destruction pathway to molecular synthesis over the whole timespan under study. With the specific high helium yield in TF01 initial elemental composition, it is thus not surprising that molecule formation is hampered in this ejecta. As a consequence, no carbon chains, starting with the synthesis of C₂, are formed and released at day 1000 owing to their destruction by He⁺, as seen in Figure 4. Our results are in total contradiction with TF01 carbon dust mass yield of ∼ 0.09 M_☉. This deviation is due to the fact that their study considers the direct conversion of the initial carbon content decreased by its depletion in forming CO at steady state, without considering the ejecta full chemistry and the overwhelming destruction of small carbon clusters by He⁺. CD09 show that a non-steady state chemistry commands the evolution of the ejecta gas phase until 1000 days after explosion, and the present results illustrate again the importance of considering the ejecta chemistry and all chemical processes responsible for dust nucleation in any treatment of dust formation in SNe.

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In summary, we suggest that the high carbon dust yields derived by TF01 for fully mixed ejecta of zero-metallicity 20 M_☉ CCSNe should be used with caution for they have been derived for questionable nucleosynthesis initial conditions and using the CNT approach.

Dust synthesis in Pop. III massive SNe is studied by SFS04. The CNT approach is used to describe the condensation of solids and SiC grains are not included among the condensates considered. The initial chemical compositions resulting from explosion nucleosynthesis for stellar progenitors with mass ranging from 140 to 260 M_☉ are those of Heger & Woosley (2002, hereafter HW02). First, we note that very low temperature profiles are obtained for the various massive progenitors considered, e.g., the 195 M_☉ progenitor ejecta has temperatures decreasing from ∼ 5400 K to ∼50 K for 100 days ⩽ t ⩽ 1000 days, a somewhat similar T variation than that found in CCSN ejecta. Second, inspection of SFS04 Figure 3 shows that the ejecta temperatures decrease with increasing progenitor masses. Low temperature profiles are questionable for PISNe, which are characterized by much larger explosion energies and ⁵⁶Ni mass synthesized in the explosion nucleosynthesis than CCSNe (HW02, UN02). As a matter of fact, solving for the radiative transfer equation taking into account the energy deposition by radioactive elements, NK03 find higher ejecta temperatures for PISNe than for CCSNe. Second, more massive PISNe are characterized by larger central temperatures, explosion energies, and synthesized ⁵⁶Ni mass than their less massive counterparts (Fryer et al. 2001; HW02; UN02), normally resulting in higher ejecta temperatures due to the conversion of part of the explosion energy in thermal energy and the deposition of energy through γ-rays production from radioactive decay. As the condensation of solids in TF01 and SFS04 formalism is assumed to start when the nucleation rates reach their maximum values, the low ejecta temperatures permit the condensation of grains as early as 150 days post-explosion where the gas densities are high enough to provide large final dust yields.

Results from SFS04 and the present modeling are summarized in Table 6. SFS04 find that the amount of CO formed in the ejecta under steady state is equal to ∼6 M_☉ and 0.01 M_☉ for the 170 M_☉ and 260 M_☉ PISNe, respectively. Their dust mixture is dominated by silica, forsterite, and AC. They derive a total dust yield of ∼39 M_☉ and ∼70 M_☉ for the 170 M_☉ and 260 M_☉ progenitors, respectively. According to Table 6, the dust synthesized by a 260 M_☉ PISN also includes 8 M_☉ of magnetite Fe₃O₄, due to the large initial iron mass yield in the model of HW02. We apply our chemical kinetic approach to the SFS04 170 M_☉ and 260 M_☉ PISN ejecta models with similar initial compositions. First, we find that a large molecular phase is synthesized in the ejecta for both models even when O₂ depletion is considered in the derivation of the dust yield upper limits. For the 170 M_☉ progenitor, the molecular phase ranges from ∼ 34 M_☉ and 51 M_☉ depending on the metal depletion, while it ranges from 33 M_☉ to 36 M_☉ for the 260 M_☉ CCSN. For both progenitors, molecules include CO, O₂, and SO, and very massive PISNe are less efficient at forming chemical species in their ejecta than less massive PISNe, owing to their hotter gas and their more important radioactivity at play, as already shown by CD09. The present dust mass upper limits are different from the dust mass yields derived by SFS04 for the two massive progenitors, although somehow similar in terms of order of magnitude. However, strong disparities exist in both the chemical composition of the dust and the epoch of its formation. For example, SFS04 form AC as the atomic carbon not trapped in CO turns into AC grains. They argue that some CO is destroyed by the γ-rays produced by ⁵⁶Co radioactive decay and produces this pool of free atomic carbon, with no quantitative assessment of this process in their ejecta. CD09 show that this CO destruction pathway is of minor importance compared to alternative routes like neutral–neutral processes or destruction by He⁺. When kinetics is considered, the entire carbon initial content is quickly included into CO, resulting in much larger CO masses than those derived from steady state, and the formation of small carbon chains is suppressed. No carbon dust can thus form in the present fully mixed ejecta of PISN models as already seen in Section 4.1.

As to other solids, we find that all the (SiO)₅ rings may be converted into silica, fayalite, forsterite, and pure silicon, or silica and pure silicon, dependent of the level of depletion of atomic Mg and Fe. Two other precursors are present in various amounts dependent of progenitor mass, AlO and (Fe)₄, leading to the formation of alumina and pure iron grains. The large initial Fe elemental yield for the 260 M_☉ progenitor leads to very large amounts of pure iron dust at t = 1000 days when Fe depletion is not considered, as seen in Table 6. SFS04 find that silica, forsterite, AC, and magnetite are the dominant dust products for both progenitor masses. Their dust chemical composition is then very different from that derived in this study. Also important is the discrepancy in the condensation sequence. SFS04 find a condensation time window of 150–250 days, dependent of their progenitor masses. Nucleation epochs in the present study are controlled by chemical kinetics, reproducing dust synthesis in the laboratory. For example, the very low ejecta temperatures considered by SFS04 imply that the (SiO)₅ clusters form at 110 days post-explosion (or at temperatures close to those found in flames) whereas (Fe)₄ are synthesized at t = 250 days and AlO forms at much later times (500 days) and lower temperatures.

In summary, the SFS04 study uses very low ejecta temperatures for PISNe and the CNT approach to describe dust condensation. While the amount of dust synthesized is comparable to the upper limits we derive for similar progenitor masses, the present study points to a totally different composition for the condensed grains, in particular the formation of pure iron grains and the lack of AC and iron oxides like magnetite. Once again, caution should be taken when using CNT model outputs into star formation and chemical evolution studies in the early universe.

Cluster abundances versus time for the 170 M_☉ progenitor are presented in Figures 5 and 6 for zones 1 and 2, respectively, while the dust mass yields ejected at 1000 days assuming Mg depletion or no depletion, as explained in Section 2.4, are summarized in Table 7. While zone 1 is characterized by a unique Si/Fe-rich chemistry, zone 2 contributes the most to the cluster budget from all zones. We see from Figure 5 that the first clusters to form at ∼400 days are iron sulfide clusters, followed by pure silicon clusters at ∼700 days and pure iron clusters at ∼800 days. The FeS molecule is quickly formed from reaction (24) when its destruction occurs via thermal fragmentation and the reverse process of reaction (24). FeS formation leads to the depletion of molecular S₂ at t > 300 days, as seen in CD09. The polymerization of FeS clusters occurs via reaction (26) at t > 400 days as thermal fragmentation remains important at early times due to the high gas temperatures. Once this process is switched off, the growth of the small cubic cluster (FeS)₄ proceeds, with no other chemical destruction channels taking place. For pure silicon and iron clusters, formation of di-silicon and di-iron chains are assumed to proceed with a similar formation mechanism characterized by the three-body reaction (27). The formation of (Si)₄ occurs at slightly earlier times than that of (Fe)₄ owing to the large amounts of available atomic silicon not locked in SiS. The high Si initial composition of zone 1 spawns the large abundances of pure Si clusters, followed by iron sulfide and pure iron clusters. As to ions, the dominant species in zone 1 is Fe⁺ which forms from the charge exchange reaction between Si⁺ and Fe, Si⁺ being formed from the Compton electron ionization of atomic Si. Fe⁺ is further destroyed by its recombination to Fe, which controls the electron population in the zone.

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In zone 2, characterized by a high Si/O initial content, the conversion of atomic Si into SiO occurs for t < 250 days as seen in Figure 6, followed by the quick polymerization of SiO ring clusters at t > 350 days (equivalent to a gas temperature of ∼ 3700 K). The further growth of (SiO)₅ keeps depleting the SiO content of the ejecta zone, resulting in large (SiO)₅ abundances. We stress here that the SiO distribution obtained in the present study and shown in Figure 6 is slightly different from that presented in CD09 for the same mass zone. This is due to the fact that we treat SiO ring polymerization up to 5 (SiO) units in this study whereas (SiO)₃ was the end product of cluster formation in CD09. The discrepancy results from the replenishment in SiO molecules during the destruction of (SiO)_3,4,5 rings, leading to a higher SiO abundance distribution with time, although the SiO abundance trend remains the same. As to other condensates, AlO forms at T > 550 days from reaction (17), once O₂ has stopped to be efficiently destroyed by thermal fractionation. Finally, pure Mg clusters are synthesized at t > 800 days when their thermal fractionation has ceased.

In Table 7, we list the results on the upper limits for dust masses formed at 1000 days post-explosion for zones 1–5 and for the two cases of total Mg depletion and no depletion. In this regard, the latter case gives almost directly the mass yields of the clusters synthesized at 1000 days. As already stated, the dominant dust formation zones are zones 2 and 3, followed by zone 1. From Table 7, we see that there exist two mass classes of condensates. The first class gathers the most abundant solids, all Si-bearing compounds, silica, forsterite, and pure silicon. Indeed, the prevalent condensate in both depletion cases is silica (SiO₂), forming essentially in zones 2 and 3. If Mg depletion is considered, forsterite (Mg₂SiO₄) also forms in these zones. In both depletion cases, pure silicon clusters are too abundant owing to the disproportionation of silicon monoxide clusters, as explained in Section 2.1.1. This first class of solids represents ∼99.5% of the dust content. The second class of condensates gathers the minor compounds, alumina, iron sulfide, and pure magnesium and iron grains. All of those form separately from the first class and their mass yields are independent of the depletion case. This class accounts for ∼0.5% of the total dust budget. It is noteworthy that neither carbon grains nor periclase (MgO) and SiC precursors form in significant amounts in the unmixed ejecta of the 170 M_☉ progenitor. Despite the presence of carbon in zones 4 and 5, these zones are characterized by a C/O ratio less than unity, a very low atomic Si content, and by large amounts of helium in zone 5. Although the formation of carbon chains is triggered by the rapid thermal conversion of CO into C₂ at high temperatures and not by slow radiative association processes, as seen in CD09, those chains never form in significant quantities due to either their overwhelming oxidation by atomic O which hinders their growth in zone 4, or the presence of He⁺ in zone 5. As to SiC precursors, they do not form owing to the small Si initial content of zone 5 and the fact that the synthesis of gas-phase SiC competes with the faster formation channels for C-chains, and suffers severe destruction by He⁺.

Upper limits of the mass yields for each solid versus depletion case are shown in Figure 7 for the present 170 M_☉ progenitor along with results from NK03 for a similar unmixed ejecta. According to Table 7, the total amount of dust formed for the 170 M_☉ progenitor ranges between 5.7 M_☉ and 7.1 M_☉. This is well below the ∼ 25 M_☉ of dust synthesized in NK03 170 M_☉ ejecta. As seen from Figure 7, our dust chemical composition also disagrees with that of NK03 who find that ∼75% of the grains are made of pure silicon, forsterite, and silica grains, when the last 25% are pure iron and iron sulfide grains, periclase, and AC, the latter accounting for ∼1% of the total dust mass.

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Cluster abundances for zones 1 and 2 of the 20 M_☉ progenitor are shown in Figures 8 and 9, respectively while upper limits for dust masses formed at 1000 days post-explosion for zones 1–4 and for the two cases of total Mg depletion and no depletion are presented in Table 8. The chemical processes at play in the formation of clusters are similar to those already highlighted in Section 4.3.1 for zones 1 and 2 of the 170 M_☉ ejecta. In Figure 8 and for zone 1, the prevalent clusters are, in order of decreasing abundances, (FeS)₄, (Si)₄, (SiO)₃, and (Fe)₄. A major departure arises from the large initial mass of oxygen compared to zone 1 of the 170 M_☉ ejecta. In this Si-rich zone, the large O mass combined with the low gas temperature fosters the early nucleation of (SiO)₃ clusters at t ∼ 250 days, while SiO clusters are absent from zone 1 of the 170 M_☉ ejecta. Moreover, zone 1 contains 8 times more Fe than zone 1 of the 170 M_☉ ejecta. All these discrepancies spawn the larger efficiency at making dust in zone 1 for the CCSN, as shown in Table 8. However, the formation of Fe-rich silicate like fayalite (Fe₂SiO₄) is hampered in zone 1 due to the fact that all the initial oxygen is trapped into the formation of SiO, leaving no oxygen for further oxidation by O₂ or atomic O. Thus, the large amounts of (SiO)₃ result in silica formation. When zone 2 forms essentially silica and/or forsterite, we see from Table 8 that zones 3 and 4 do not contribute to the cluster budget. Zone 3 is Si-poor with a C/O ratio less than unity, implying that CO is the prevalent species to form in this zone (CD09). Zone 3 is thus not conducive to cluster synthesis. Zone 4 is characterized by a large C/O ratio (29.5) and a high He content. These conditions preclude the synthesis of C-chains and rings and the subsequent AC grain condensation because of the destruction of small clusters by He⁺, as explained in Section 4.4.2. For SiC precursors, conclusions similar to those drawn for the 170 M_☉ case apply, and no silicon carbide dust forms in zones 3 and 4.

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Upper limits of the mass yields for each solid versus depletion case are shown in Figure 10 along with the results from NK03 for a similar unmixed ejecta. According to Table 8, the total amount of dust formed for the 20 M_☉ progenitor ranges between 0.103 M_☉ and 0.154 M_☉. This is well below the ∼ 0.58 M_☉ of dust synthesized in NK03 170 M_☉ ejecta. As seen from Figure 10, our dust chemical composition disagrees with that of NK03 who find that ∼64% of the dust mass is in the form of pure silicon, forsterite, and silica grains, 16% and 5% are iron sulfide and pure iron grains, respectively, and 10% is in the form of AC.

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Elements initially in atomic form are first depleted in molecules and molecular clusters as a result of nucleation chemistry. To assess this trapping independent of the depletion cases defined in Section 2.5, we define the efficiency x as

$$\begin{equation} x = N_{\rm Cluster}(y) / N_{\rm Total}(y), \end{equation} \tag{ 41 }$$

where y is the element, N_Cluster(y) is the total number of y atoms locked up in one type of cluster, and N_Total(y) is the initial total number of y atoms in the ejecta. The efficiency x measures how the chemistry locks up elements in the nucleation processes of molecular clusters. It is not linked to the potential depletion of Mg and Fe during the condensation phase of dust seeds. Depletion efficiencies are shown as a function of zoning in Table 9 for the two progenitor masses.

For both progenitor masses, the depletion of Si atoms into silicon oxide precursors leading to silica and pure silicon clusters is efficient in silicate forming zones and starts at early times when Si is converted into SiO and small SiO clusters (see Figures 6 and 9). The depletion is then total at day 1000 as seen in Table 9 and Sections 4.3.1 and 4.3.2. This is in agreement with the steeper fading of the 1.65 μm [Si i] line observed in SN1987A compared to the 4571 Å [Mg i] and the 6300 Å [O i] lines observed by Lucy et al. (1989). In the Fe-rich, O-poor zone 1, the conversion of Si into pure silicon clusters proceeds with a low efficiency of ∼1%–3% due to the prevalent conversion of Si in SiS molecules with an efficiency ranging from 26% to 28%.

Aluminum also suffers total depletion into molecular AlO. The formation of alumina clusters has not been modeled here but AlO should react with itself and O₂ to build up small Al₂O₃ clusters (Catoire et al. 2003). A total depletion of Al into alumina clusters is thus to be expected.

Carbon is massively depleted into CO in the zones where it is present. In the C-rich zones 4 and 5 of the 20 M_☉ and 170 M_☉ ejecta, respectively, its depletion depends on the presence of He⁺, as fully explained in Section 4.4.2. In zone 5 of the 170 M_☉ PISN, which has a C/O ratio of 0.66, most of the carbon is tied up in CO once the He⁺ abundance decreases for t > 600 days (see Figure 7 of CD09). Conversely, in zone 4 of the 20 M_☉ CCSN characterized by a C/O ratio of 29.5, C remains in atomic form due to the overabundance of He, hence He⁺. However, atomic carbon is almost entirely depleted in the carbon ring C₁₀ if He is not microscopically mixed to the gas, as seen in Section 4.4.2.

Atomic oxygen is depleted in molecular O₂ with efficiency values ranging from 12% to 61% in all zones where O is present, and into CO (x ≡ 29%–43%) where atomic carbon is abundant. The formation of silica clusters locks up atomic oxygen to efficiency values spanning ∼3%–10% only. Thus part of the decline in the oxygen lines is due to molecular formation rather than dust condensation.

Iron is depleted in the formation of FeS clusters in zone 1 of both ejecta. Its depletion efficiency in FeS reaches the high level of 44% for the 20 M_☉ progenitor ejecta. We also see that Fe is never efficiently transformed into Fe clusters. To assess if this low efficiency in nucleating iron small clusters is due to the presence of sulfur and the formation of FeS, we ran the chemistry for the S-free zone 1 of the 20 M_☉ progenitor. We find that the depletion of iron into small Fe clusters stays very low (∼0.3%) and that most of the iron remains in atomic form. As to Mg, it is seldom depleted by the nucleation of small clusters except in zones 3 and 2 for the 170 and 20 M_☉ ejecta, respectively, with depletion efficiencies of 1% and 2%. However, the coagulation into large silica clusters may trap atomic Mg or Fe into the cluster lattices to form forsterite and fayalite, respectively, a situation described by our "total depletion" case defined in Section 2.1. However, these depletion processes are beyond the aims of this paper and will be presented in a forthcoming publication.

A critical parameter to the formation of silicates and metal oxides is the rate at which small clusters react one to another to nucleate small dust seeds that can later coagulate and condense into larger dust grains. In previous chapters, we assume that reactions (7), (12), (16), (23), and (26) all proceed with similar rates listed in Table 2, as the only information available on nucleation rates is for small silicon oxide clusters (ZT93). To test the impact of varying the nucleation rates of small clusters of silicon oxide, metal oxides, and sulfides, we consider a case study describing the nucleation of small silicon oxide clusters in zone 2 of the 170 M_☉ progenitor unmixed ejecta. The rates for reactions such as reactions (7) are listed in Table 2 and are taken from the ZT93 study for the 10⁻² atm pressure case. Using this series of rates as benchmarks, we now study the impact of increasing and decreasing by a factor of 100 the benchmark rates. The increase by 100 corresponds to the 1 atm pressure case of ZT93 and we calculate the abundances normalized to total gas number density of clusters as a function of post-explosion time.

Abundances for the (SiO)₅ clusters and gas-phase SiO are presented in Figure 11 for the different rates of reaction (7). As the nucleation rate decreases (e.g., in the 10⁻⁴ case for which all rates have been decreased by a factor of 10⁻² compared to our benchmark case), the formation of clusters is postponed to late times while the SiO abundance increases first and starts dropping once (SiO)_n clusters form. However, the abundance of the (SiO)₅ cluster is unchanged at day 1000. Though not shown in Figure 11, the O₂ and CO abundances follow similar trends, implying that the lower the rates, the larger the molecular phase and its mass in the ejecta, while the final cluster mass yields at day 1000 are similar for all cases. As seen in Section 4.3.3, elemental Si and SiO are totally depleted in the process of silica and silicate dust formation. Therefore, observations of Si and SiO optical and IR lines in local young SNe are good indicators of the timing and the efficiency of the nucleation of silicate dust in SN ejecta, once excitation effects are taken into account in the analysis of line declines as a function of time.

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No carbon rings form in the unmixed ejecta of the 170 M_☉ and 20 M_☉ progenitors owing to the large amounts of He and hence He⁺ in C-rich zones 5 and 4, respectively. However, evidence for the presence of carbon grains in CCSNe ejecta exists from various sources of investigation. Graphite grains bearing the isotopic anomalies of CCSNe are found in meteorites (Zinner 2006). Furthermore, models aiming at explaining the IR excess emission of several low-z CCSNe and SN remnants show that AC is a necessary component of the dust mixture required to reproduce IR data (Sugerman et al. 2006; Ercolano et al. 2007; Rho et al. 2008). In order to assess the impact of He/He⁺ on the formation of carbon chains and rings, we re-investigate the chemistry of zone 4 of our 20 M_☉ progenitor, which is characterized by a C/O ratio of 29.47. Keeping the initial metal mass constant, we gradually increase the amount of He present in the zone up to its value assumed in the previous section (0.884 M_☉ according to Table 3 of CD09). We consider the following increase factors: 0% (≡ no He), 1%, 20%, 40%, and finally 100% of the He value of zone 4. For increasing He masses, the initial total number of elements in the gas also increases while the partial number of metals is kept constant. Doing so, we aim at reproducing the chemical composition of a C-rich clump in the ejecta partially embedded in a helium mushroom cap, a configuration commonly found in two-dimensional/three-dimensional modeling of CCSN explosion (Müller et al. 1991; Kifonidis et al. 2003; Hammer et al. 2009). We then allow some diffusion of He inside the C-rich clump.

The abundance of the ring C₁₀ as a function of post-explosion time for our various He diffusion values is shown in Figure 12. We see that the larger the He content of the zone, the later the formation and the smaller the abundances of C₁₀ rings. Indeed, for very large He content, the amount of He⁺ in the zone is large (x(He⁺) ∼ 0.1% × x(He), where x(He) and x(He⁺) are the abundances of He and He⁺, respectively). The destruction of small carbon chains like C₂ then proceeds until He⁺ abundances diminish at late times owing to a decrease in He ionization by Compton electrons and the recombination of He⁺. The resulting chain and ring formation is thus delayed. When the zone is free of He, the rapid conversion of the atomic carbon not locked in CO into C₂ via the radiative association reaction (30) triggers the formation of larger chains, with instantaneous closure of the end-product ring C₁₀. For the He-free zone 4, the final abundance of C₁₀ with respect to total gas number density at 1000 days is 0.55, equivalent to an AC dust mass yield of 0.0145 M_☉  in which ∼95% of the initial carbon is depleted. When He diffusion occurs, the conversion of free atomic carbon into C₁₀ is as effective as for the He-free case, with similar final mass yield and formation efficiency for the C₁₀ ring, except that the conversion takes place at later times until the He⁺ abundance decreases.

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In summary, the build-up of carbon clusters in SN ejecta is significantly hampered by oxidation in zones where the C/O ratio is less than unity, and by the microscopic mixing of He⁺ in the carbon-rich zones of the ejecta. More generally, He⁺ has a detrimental impact on molecular formation in SN ejecta as first shown by Lepp et al. (1990) and CD09. In C-rich, He-free zones, carbon condenses with a high efficiency. Any formation of carbon dust in SNe must occur in C-rich inhomogeneities deprived of He⁺ and is thus highly dependent of the mixing induced by the SN explosion.