GoKawiilKineCond is a code developed to calculate the time-dependent condensation processes of minerals in the solar nebula, which is dominated by hydrogen. It is described in detail in Supplementary Section 3 with several validation tests, and here we give a summarized version. The elements considered in the system are H, He,O, Mg, Si, Fe, Al, Na, K, Ni, Ca, Cr, S and C. Here, the term element designates an atomic species. The system consists of a gas in interaction with 39 minerals. Most of the condensation codes published in the past and that allowed the investigation of the Equilibrium Condensation Sequence3,4,5,7 rely on the Gibbs free-energy minimization (GFEM). GFEM calculates—for a given pressure and temperature—the most stable combination of minerals and gas; however, like any equilibrium calculation, it provides no information if the equilibrium state is established in a reasonable time, nor does it give the list of reactions through which the equilibrium state is realized (even though some additional physical arguments may help to determine those reactions, especially at high temperature when the number of minerals present is small). However, each reaction has its own kinetics, which depends on pressure, temperature and the local abundances of all elements. Thus, to design a time-dependent kinetic code, we must adopt a different strategy. We must explicitly specify the list of all reactions of interest, and advance each of them individually in a fully coupled way.
KineCond proceeds as follows: at each time step, we compute the number of atoms in the gas and the number of atoms in every mineral, keeping the total number of atoms constant. We assume that the pressure is constant and that only the temperature varies with time. The evolving variables of the system are: the number of moles of each molecule i in the gas (N i gas) and the number of moles of each mineral j in the system (N j min; Supplementary Table 2). We assume that gas–gas reactions are much faster than gas–mineral reactions and condensation reactions, so the gas molecular composition is always close to chemical equilibrium (then the gas molecular composition only depends on the current values of temperature and pressure, as well as N i gas). As we focus on gas–mineral processes, reactions are divided into two broad categories: (1) condensation or evaporation reactions; and (2) gas–mineral reactions. The rates of these reactions dictate the evolution of N i gas and N j min. The temperature varies linearly with time, dropping from 2,000 K to 130 K on the timescale T c (ranging from 0.01 to 2000 years). The system evolves as follows: at each time t we first calculate the molecular composition of the gas (excluding mineral condensation) by computing the chemical equilibrium of the gas at T(t) and P and with elemental abundances N i gas(t). This is performed using the iconic Chemical Equilibrium with Application distributed by NASA (CEA-NASA) code47, which includes about 1,500 gas species in total. The instantaneous gas molecular composition is then used to compute the different condensation and gas–mineral reactions and the rate at which they proceed. The different steps of the calculation are summarized in a flow chart provided in Supplementary Fig. 5. We detail these calculations below.
Condensation or evaporation reactions We follow a condensation and evaporation theory developed for forsterite evaporation in a H 2 gas48,49, and we generalize it to many minerals. The net formation rate of a mineral j is the difference between an evaporation flux (J j e) and a condensation flux (J j c). Each of them must be computed explicitly. In the gas, the flux of any element E across a unit surface (in moles m–2 s–1) is calculated using the kinetic theory of gases48,49:
$${J}^{c}(E)=\sum _{m}\frac{{
u }_{m}^{E}{P}_{m}}{({2{\rm{\pi }}{\mu }_{{\rm{m}}}RT)}^{1/2}}$$ (1)
where m is any gas molecule; \({
u }_{m}^{E}\) is the stoichiometric coefficient of element E in molecule m; P m is the partial pressure of molecule m; µ m the molar mass; and R is the ideal gas constant. The partial pressures of gas molecules (P m ) are obtained by running the CEA-NASA code. We now consider a mineral j with formula \(\{{\alpha }_{j}^{E}E\}\), where \({\alpha }_{j}^{E}\) is the stoichiometric coefficient of element E in mineral j. The condensation flux of the mineral j (J j c) is determined by the smallest flux (over all elements E entering in its composition) so that:
$${J}_{j}^{{\rm{c}}}={\gamma }_{E,j}{\min }_{E}\left(\frac{{J}^{{\rm{c}}}(E)}{{\alpha }_{j}^{E}}\right)$$ (2)
γ E,j is the sticking efficiency of atom E on mineral j. It ranges from 0 to 1 and is poorly constrained.
Here we set γ E,j = γ = 0.1, as a standard value, for all minerals and all atomic elements.
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