Astron. Astrophys. 332, 1099-1122 (1998)

4. Equilibrium chemistry of the Mg-Fe-complex

The possible condensates formed from the elements Mg, Fe, and Si are only stable for temperatures where (i) H is completely associated to H₂, (ii) carbon not bound into carbon dust has formed CO and is completely locked up in this molecule, (iii) gas phase silicon is completely bound in the SiO molecule, and (iv) the remaining fraction of gas phase oxygen not bound in SiO or CO has formed H₂ O molecules. The metals Fe and Mg are present as free atoms in the gas phase and nitrogen is completely bound in N₂. A lot of much less abundant additional molecular species of these elements, and atoms or molecules of the less abundant elements exist in the gas phase, but these are of no interest in the present context. At temperatures below K when Si, Mg, and Fe are condensed into solids the SiO and CO are converted into SiO₂ and CO₂, respectively, and the metal atoms form hydroxides, but this not important for our calculations.

In the following we consider the chemical equilibrium between a gas phase which is essentially composed of H₂, He, H₂ O, CO, SiO, Fe, and Mg and the possible condensates which may be formed from these elements.

4.1. Pure metals

Let P denote the pressure of the gas phase and let P_H be the fictitious pressure of all hydrogen nuclei if they were present as free particles. If the hydrogen is completely associated to H₂ we have

is the abundance of He (relative to H). Consider the condensation of metallic iron from the gas phase and let f be the fraction of the totally available iron condensed into solid iron particles. If Fe is not bound into any other condensate the fraction of the Fe remains in the gas phase. Its partial pressure then is

In order that the condensed iron metal and the free iron atoms in the gas phase are in a state of thermodynamic equilibrium, the partial pressure of Fe atoms has to be equal to the vapour pressure of Fe atoms over solid iron, which is given by (law of mass-action ¹)

denotes here and in all of the following equations the Gibbs free enthalpy of formation of the substance under consideration (solid iron in the present case) from free atoms. From (9) and (10) we get

This uniquely defines, for any given temperature T, the total pressure P in a state where just the fraction f of the total available iron has condensed into solid iron. defines the limit curve in the P -T plane above and to the left of which no condensed iron can exist. We calculate the position of this limit curve using thermodynamic data for from Sharp & Huebner (1990). If not explicitly stated otherwise, in all of the following calculations thermodynamic data are taken from this source. Element abundances are that given by Anders & Grevesse (1989) with the updatings of Grevesse & Noels (1993). The result is shown in Fig. 3. Obviously, metallic iron is one of the most stable dust components in the disk which can be formed from the most abundant refractory elements.

Fig. 3. Limit curves for the formation of refractory condensates of the abundant elements Mg, Fe, and Si from atoms or molecules from the gas phase. The dashed line shows the P -T stratification in the central plane of a protoplanetary disk model. The upper and lower inflections in this line correspond to the stability limits of corundum and olivine, respectively. The dotted line shows the limit curve for H2 dissociation for comparison

The iron easely forms alloys with some other elements, especially with the somewhat less abundant Ni. Such solid solutions are not considered in this paper.

The limit curve for vapourisation of solid Mg is calculated in the same way as for iron and the result also is shown in Fig. 3. Obviously, no solid magnesium particles can exist in the disk.

4.2. Siliconoxide

Silicon forms the solid oxide SiO₂. This may be formed from the gas phase by for instance the reaction

The solid and the gas phase species are in chemical equilibrium if the partial pressures of the molecules satisfy

where . Let f be the fraction of the totally available silicon bound into SiO₂. If no other condensed silicon compound is present, then the partial pressure of SiO molecules in the gas phase is and if no oxygen is bound in any other one condensate, the partial pressure of H₂ O in the gas phase is . It follows

This, again, determines for any T the total pressure P of the gas phase in that state where just the fraction f of the silicon is condensed into SiO₂. The limit curve for stability of SiO₂ corresponds to . This limit curve is shown in Fig. 3 for the temperature and pressure region of interest for protoplanetary accretion disks. Since the stability limit for SiO₂ is below and to the right of the stability limit of the Mg-Fe silicates (cf. Fig. 3) to be discussed later the SiO₂ is not expected to exist in a chemical equilibrium state.

The relation (13) does not depend on the special choice of a reaction considered for the formation of the SiO₂. Since we consider a thermodynamic equilibrium state, the principle of detailed balance holds according to which the equilibrium state does not depend on the special process responsible for its formation. Assuming any other reaction for the formation of SiO₂ would yield just the same limit curve. Thus we are free to choose in the present case, as well as in all of the following considerations, that reaction for the formation of a substance which is the most convenient one for calculational purposes.

The method used here and in the following considerations for calculating abundances of solid compounds is not as flexible as the method of minimization of the total Gibbs free energy of the system applied for calculating the abundances in a mixture of a big number of molecules, solid compounds, and solid solutions as implemented, for instance, in the codes used by Saxena & Eriksson (1986) or Sharp & Huebner (1990), but the approximate formulas to be developed in this paper are well suited for computational purposes if combined with semi analytic equations for the disk structure like that given in Sect.  6.1.

The molecule SiO alternatively can condense into the less stable solid siliconmonoxide. This may be of interest for circumstellar dust condensation (e.g. Gail & Sedlmayr 1998) but is of no interest for accretion disks.

4.3. Iron sulphide

One important component of the dust mixture seems to be solid FeS. We consider its formation by a reaction between solid Fe and S molecules from the gas phase

In chemical equilibrium the partial pressures of the molecules have to satisfy

where . If we assume that the fraction f of the totally available iron has reacted to FeS then we have for the partial pressure of H₂ S in the gas phase

From (14) it follows for a chemical equilibrium state

Here the pressure P of the gas phase drops from the expression. The solution for f of this equation

only depends on the temperature. The maximum fraction of the iron which can be converted into FeS equals the abundance ratio . If the quantity f calculated from Eq. (16) for a certain temperature T takes a value in the interval , then for this temperature the iron metal is partially converted into solid FeS. If there results a value of , then all the iron is present as the free metal and we have to put in this case. Fig. 4 shows the result for f for solar system element abundances. An inspection of the figure shows that the conversion of FeS to Fe is extended over a broad temperature region. The reduction of FeS starts to become significant at 500 K, gradually increases with increasing T and is completed at a temperature of K. Above this temperature, the iron would be present as the free metal and no FeS can exist at such temperatures due to reduction of FeS by the abundant hydrogen. Below this temperature the iron would form FeS due to sulphidisation by H₂ S, but this cannot convert all of the iron into FeS since S is less abundant than Fe. If iron is not bound in some other solid, the excess over S would be converted below 370 K into iron oxide. The kinetics of the formation of FeS has been studied in detail by Lauretta et al. (1996).

Fig. 4. Fraction f of metallic iron which is converted by H2 S into FeS (troilite) in matter with solar system element abundance.

4.4. Iron oxides and hydrides

Iron and magnesium may form oxides and hydroxides. Such compounds occur only at temperatures well below 400 K (e.g. Fegley 1989) and are of no interest for the warm inner parts of an accretion disk where the main dust materials are destroyed. The iron oxides and hydroxides will be considered in a separate paper on iron grains in accretion disks (Finocchi & Gail 1998).

4.5. Equilibrium chemistry of the Mg-Fe ortho-silicates

The dust model of Pollak et al. (1994) considers two types of magnesium-iron silicates:

• The ortho silicates with the composition . They form a continuous series of compounds with . The two members at the endpoints of this series are fayalite () and forsterite (). The intermediate case is known as olivine. In the P94 model is assumed.
• The meta-silicates with the composition . They form a continuous series of compounds with . The two members at the endpoints of this series are ferrosilite () and enstatite (). The intermediate case is known as orthopyroxene. In the P94 model is assumed.

First we consider the ortho-silicates. We determine the limit curves for the stability of the extreme members of this series. The law of mass action for the reaction of formation of forsterite (Mg₂ SiO₄) from gas phase species according to the reaction

is

where . Let f be the fraction of the silicon bound in Mg₂ SiO₄. If there exists no other dust condensate than this one, the partial pressure of SiO in the gas phase is , that of Mg is and that of water vapour is . Inserting this into (17) yields

The limit curve for complete destruction of Mg₂ SiO₄ is shown in Fig. 3. If Mg is replaced by Fe in this equation, one obtains the corresponding equation for fayalite (Fe₂ SiO₄). The limit curve for complete destruction of Fe₂ SiO₄ by volatilisation into gaseous components is shown in Fig. 3, too. We observe that forsterite is much more stable than fayalite.

Next we consider the magnesium-iron silicates (olivines) with a mixed composition with . We assume that the members of this series form a solid solution of forsterite and fayalite. The free enthalpy of formation of one mole of the mixture from x moles of forsterite and moles of fayalite is given by the weighted mean of the free enthalpy of formation of both components and the mixing entropy term

(Atkins 1994). We consider the conversion of forsterite into olivine by means of the reaction

In a chemical equilibrium state between the gas phase and olivine, the partial pressures of Mg and Fe atoms in the gas phase satisfy the law of mass action

where is the change of free enthalpy in the conversion of forsterite into olivine. It follows

where we denote the equilibrium constant for the conversion by . This equation determines the pressure ratio of Mg and Fe atoms in a chemical equilibrium state with olivine. For the equilibrium constant we have

where

The temperature dependence of for some values of x is shown in Fig. 5.

Fig. 5. Equilibrium constant for the conversion of forsterite into olivine. The values of x are indicated at the curves

If f is the fraction of silicon bound in the olivine, the partial pressures of Mg and Fe in the gas phase are and , respectively. Then

The degree of condensation f is determined by Eq. (18) where the term has to be replaced by . Both Eqs. (24) and (18), together determine the amount and composition of a magnesium-iron silicate condensed in a chemical equilibrium state. We can solve (24) for f

For instance, if then inspection of Fig. 5 shows and then , as is to be expected from solar system element abundances. Complete destruction of the silicate means which means according to (25) (element abundances according to Anders & Grevesse (1989)). An inspection of Fig. 5 shows that in this case . At the stability limit the material has nearly the composition of pure forsterite. The reason for this is simple: at sufficiently high temperature the fayalite component is distilled off from the solution.

For given x and T Eq. (25) determines f and then from Eq.  (18) we can determine the total gas pressure corresponding to this state. Equilibrium curves in the P -T plane for some interesting values of x are shown in Fig. 6. An inspection of the figure shows that in a broad temperature region below the limit for decomposition of the olivine the chemical equilibrium composition of olivine would be that of nearly pure forsterite. Only at temperatures at least 200 K below the stability limit would substantial amounts of Fe be incorporated into the silicate. Inspection of Fig. 3 or Fig. 6 shows that the iron condenses as the free iron metal at similar temperatures as the olivine and hence in a broad region the chemical equilibrium composition of the condensate corresponds to a mixture of nearly pure forsterite and metallic iron particles!

Fig. 6. Limit curves for conversion of forsterite into olivine. The numbers denote the value of the stoichiometric coefficient x for olivine with composition . The uppermost full line is the stability limit of pure forsterite (). The upper dotted line shows the stability limit of enstatite, the lower one where 90% of the forsterite is converted into enstatite. The dashed line shows the P -T stratification in the central plane of a protoplanetary disk model for comparison.

In constructing Fig. 6 we have assumed that iron is not condensed. The formation of solid iron would reduce the pressure of Fe atoms in the gas phase and this would favour a value of x even closer to unity.

4.6. The Mg-Fe meta-silicates

The same type of consideration yields for the curve of constant degree f of condensation of silicon into the meta-silicate enstatite

and the corresponding equation for ferrosilite with Mg replaced by Fe. The limit curve for complete destruction of MgSiO₃ is shown in Fig. 3. It is nearly as stable as Mg₂ SiO₄ and is destroyed at an only slightly lower temperature than forsterite. This means, that there exists the possibility that ortho- and meta-silicates may coexist in a chemical equilibrium state. We shall come back to this point below and consider first the conversion of enstatite into orthopyroxene.

We consider the magnesium-iron silicates with a mixed composition with . They are assumed to form an ideal solid solution of ferrosilite () and enstatite (). The free enthalpy of formation of one mole of the mixture from x moles of enstatite and moles of ferrosilite is given by

Now, we consider the conversion of enstatite into orthopyroxene by means of the reaction

In chemical equilibrium the partial pressures of Mg and Fe atoms in the gas phase satisfy

where is the change of free enthalpy in the conversion of enstatite into orthopyroxene. It follows

where again we denote the equilibrium constant for the conversion by . For the equilibrium constant we obtain

where

The result for for some values of x are shown in Fig. 7. Data for FeSiO₃ are taken from Saxena & Eriksson (1986). is very small since the conversion of MgSiO₃ into FeSiO₃ is strongly endothermic with kcal/ Mol.

Fig. 7. Equilibrium constant for the conversion of enstatite into orthopyroxene. The values of x are indicated at the curves

If f denotes the fraction of silicon bound in orthopyroxene, the partial pressures of Mg and Fe in the gas phase are and , respectively. Then

Since for the relevant temperatures is of the order of this requires . Since x and f by definition both are less than unity and for a solar system like element composition there exists no solution of (32) for x if only the meta-silicate would exist. In this case only pure enstatite exists in chemical equilibrium. If part of the Mg is bound in ortho-silicates, the orthopyroxenes with may exist.

Next we consider the possibility of a coexistence of ortho- and meta-silicates. Since the iron content of the silicates in any case is small, as we have seen, we can restrict our considerations to pure Mg-silicates. As can be seen from Fig. 3 the ortho-silicate is stable up to a slightly higher temperature than the meta-silicate. In a chemical equilibrium state with the hydrogen rich gas phase, coexistence between ortho- and meta-silicates means

According to the law of mass-action the partial pressures of the gaseous species have to satisfy in equilibrium

where . Assuming that no other dust species are present than the two species presently under consideration the partial pressure of Mg in the gas phase is , the partial pressure of H₂ O is , and the partial pressure of H₂ is . and denote the fraction of the silicon condensed in the ortho- and meta-silicate, respectively. It follows

This is the total pressure P of the gas phase in a chemical equilibrium state between the ortho- and meta-silicate for given , , and T. The limit where enstatite just starts to be converted into enstatite is defined by . Additionally we have Eq. (18) for the equilibrium between forsterite and the gas phase. Adapted to the present case it reads as follows

Equating this to (35) yields the equation

for in terms of or vice versa. This relation only depends on the temperature T but not on the pressure P.

Let and solve for :

Using this in (36) one obtains for given T the total pressure P in an equilibrium state where the abundance of enstatite is a given multiple z of the abundance of forsterite. At the stability limit of enstatite we have . This defines a limit curve in the P -T plane, above and to the left of which only forsterite exists in a chemical equilibrium state and below and to the right of which forsterite and enstatite both coexist. Fig. 8 shows this limit curve for the pressure and temperature conditions of interest for the protoplanetary accretion disk. The figure also shows the curve with where most (90%) of the condensed silicate in chemical equilibrium forms enstatite. Thus, only in a rather narrow strip of the P -T plane the forsterite forms a significant fraction of the mixture of silicates. At low temperatures the enstatite dominates in chemical equilibrium and the abundance of the forsterite component decreases gradually with decreasing temperature. It does not drop, however, below an abundance of .

Fig. 8. Equilibrium between forsterite (Mg2 SiO4) and enstatite (MgSiO3) and the limit curves for stability of forsterite and solid iron. The dashed line shows the P -T stratification in the central plane of a protoplanetary disk model.

4.7. Iron-magnesium-silicon compounds in the disk

According to our above findings the following compounds of Si, Fe, and Mg are formed in chemical equilibrium:

1. Olivine and Orthopyroxene. At low temperatures the magnesium is consumed by the formation of olivine and orthopyroxene. The orthopyroxene is the more abundant of the two magnesium silicates up to temperatures close below the stability limit for the conversion of orthopyroxene into olivine. Above this stability limit the magnesium forms olivine up to the stability limit of this compound. The iron content of olivine and orthopyroxene is small, especially at elevated temperatures. Hence olivine in an environment with Solar System abundances in fact is nearly pure forsterite and orthopyroxene is nearly pure enstatite.
2. Iron. The orthopyroxene and olivine consume only a small fraction of the available Fe. The excess of the iron is condensed into pure iron particles. The stability limit of condensed iron crosses the stability limit of forsterite. At low pressures condensed forsterite exists up to slightly higher temperatures than iron while for high pressures iron exists up to slightly higher temperatures then olivine.
3. Troilite. At low temperatures the iron forms with the available S solid FeS (troilite). Since the Fe element abundance exceeds the S abundance, the excess of Fe over S forms pure iron particles.

As our discussion of the equilibrium abundances of Si,Fe, and Mg compounds shows, these compounds occur in any environment in the above order of temperature. Only the precise value of the temperatures where the different compounds appear or disappear depend on the details of the P -T stratification in the accretion disk.

© European Southern Observatory (ESO) 1998

Online publication: March 30, 1998
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