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

5. Equilibrium chemistry of the Al-Ca complex

Aluminium and aluminium-calcium compounds tend to be more stable than magnesium-iron silicates. For this reason there exists a certain zone in the inner part of the accretion disk where no silicate dust exists but the refractory aluminium compounds do exist and dominate the extinction of the disk material and, by this, the disk structure. From the results of the calculations of Grossman (1972), Saxena & Eriksson (1986), and Sharp & Huebner (1990) for the equilibrium composition of an oxygen rich mixture we can see, that five different abundant aluminium and calcium compounds can be formed from a solar system element mixture which may be important for the disk structure: (i) aluminium oxide or corundum (Al₂ O₃), (ii) melilite, a solid solution of gehlenite (Al₂ Ca₂ SiO₇) and åkermanite (MgSiCa₂ SiO₇), (iii) spinel (MgAl₂ O₄), (iv) diopside (CaMgSi₂ O₆), and (v) anorthite (CaAl₂ Si₂ O₈).

5.1. Aluminium oxide

The most refractory of these compounds is the solid aluminium oxide due to its extremely high energy of formation. Molecules formed from Al and O, on the other hand, do not have a particular high bond energy. At such temperatures where the Al₂ O₃ disappears, the particle densities of the aluminium bearing molecules AlO and AlO₂ in the gas phase are small compared to the particle density of the free Al atoms. The oxygen at the same temperature is bound mainly in H₂ O, CO, and SiO. Since CO and SiO are not involved in the chemistry of Al₂ O₃ formation or destruction, we assume that the Al₂ O₃ is formed by the reaction

from gas phase species. In a state of thermodynamic equilibrium between the molecules involved in this reaction and the solid, the partial pressures of the gas phase species satisfy the law of mass action

where . Assuming that the fraction f of the aluminium is condensed into corundum the partial pressure of free Al atoms in the gas phase is and if no other oxygen bearing condensate than the Al₂ O₃ is present the partial pressure of H₂ O in the gas phase is . From (39) we obtain

Fig. 10 shows the resulting curves for some values of f for pressure and temperature conditions of interest for protoplanetary accretion disks. corresponds to the limit curve left and above of which no condensed Al₂ O₃ exists. A comparison with Fig. 3 or Fig. 9 shows that the Al₂ O₃ indeed is stable up to a much higher temperature than forsterite, the most refractory of the silicates. There exist a region K wide in temperature where corundum is stable but not any one of the silicates.

Fig. 9. Stability limits of solid aluminium and aluminium-calcium compounds against volatilisation and stability limits for conversion of some aluminium and aluminium-calcium compounds. The dashed lines show the dissociation limits of H2, the pressure-temperature stratification in the midplane of the accretion disk model, and the stability limit of the most refractory of the silicates, the forsterite (Mg2 SiO4).

Fig. 10. Equilibrium condensation of corundum (Al2 O3). The numbers denote the fraction f of the aluminium condensed into Al2 O3. The dotted line shows the dissociation limit of H2 and the dashed line the pressure-temperature stratification in the midplane of the accretion disk model. The upper and lower inflections in this line correspond to the stability limits of corundum and olivine, respectively.

5.2. Melilite

The most stable Al-Ca and Ca-Mg silicates at high temperature are gehlenite (Ca₂ Al₂ SiO₇) and åkermanite, (Ca₂ MgSi₂ O₇) which form a solid solution called melilite. Since the åkermanite has only a small concentration in the mixture (Saxena & Eriksson 1983) we neglect in our calculation this component of melilite.

Consider the hypothetical case that no other aluminium bearing condensate then gehlenite is present. At the relevant temperatures the gas phase species available to form gehlenite are free Ca and Al atoms, and SiO and H₂ O molecules. The gehlenite then may be formed by the reaction

According to the law of mass action the partial pressures of the molecular species in the gas phase have to satisfy

where . If a fraction g of the calcium is bound in gehlenite, we have for the partial pressures of the free atoms , , and for the molecules , . It follows

The curve with corresponds to the stability limit of gehlenite. As can be seen from Fig. 9 at the upper stability limit for volatilisation of gehlenite the corundum is stable. The aluminium liberated by the volatilisation of gehlenite then forms solid corundum. Thus, gehlenite does not disappear by decomposition into gaseous species as in reaction (41) but it is converted into corundum instead.

The relevant reaction for formation of gehlenite from corundum (or vice versa) is

The partial pressures of the gaseous species in a chemical equilibrium state between gehlenite and corundum have to satisfy

where . If the fraction g of the calcium is condensed into gehlenite and the fraction f of the aluminium into corundum, we have for the partial pressures of the gas phase species and . The partial pressures of SiO and Ca are the same as in the previous case. We obtain from (43)

The resulting curves in the P -T plane for some values of g are shown in Fig. 11. again determines the upper limit curve for stability of melilite. This stability limit is also shown in Fig. 9. There exists a strip in the P -T plane with K width in temperature in which corundum is the most stable aluminium compound. Below this at lower temperatures part of the corundum is converted into gehlenite.

Fig. 11. Equilibrium condensation of gehlenite (Ca2 Al2 SiO7). The numbers denote the fraction g of the calcium condensed into gehlenite. The dotted line shows the dissociation limit of H2 and the dashed line the pressure-temperature stratification in the midplane of the accretion disk model. The upper and lower inflections in this line correspond to the stability limits of corundum and olivine, respectively.

The fraction f of aluminium condensed into corundum in the presence of gehlenite is given by (40) which now reads as

where denotes the equilibrium constant of corundum. Neglecting the small fraction of the oxygen bound in melilite and corundum and combining the two equations for P we obtain

Eqs. (44) and (45) determine the mixture of corundum and gehlenite in a thermodynamic equilibrium state. An inspection of Figs. 9 and 10shows that the condensation of aluminium into corundum is nearly complete where conversion of corundum into gehlenite is thermodynamically favourable.

5.3. Spinel

Next we consider the condensation or disappearance of spinel (MgAl₂ O₄). First we consider the hypothetical case that no other aluminium condensate than spinel is present. Since the Mg required for the formation of spinel is present in the gas phase as the free atom, we consider the reaction

In thermodynamic equilibrium between the solid and the gas phase species we have

where . If f denotes the fraction of the aluminium condensed into spinel, the particle densities of the relevant gas phase species are , and . We obtain from (47)

defines the upper limit curve of stability of spinel in the P -T plane. This stability limit is shown in Fig. 9 which also shows the stability limit of the corundum.

Since the stability limit for formation of spinel from gas phase species occurs at a somewhat lower temperature than that for formation of corundum, the formation or disappearance of spinel does not occur by chemisputtering in a reaction like (46) but by conversion of corundum into spinel or vice versa. A possible reaction for this process is

In thermodynamic equilibrium between corundum, spinel and the species in the gas phase we have for the partial pressures of the molecules

where . Assume that a fraction f of the aluminium first is condensed into corundum and that a fraction x of this is converted into spinel. The partial pressures of the gas phase species are and . We obtain from (49)

This equation and Eq. (40) which reads in the present case as

for given T form a system of two equations for the three unknown quantities P, f, and x. They determine, thus, for fixed f or x a family of curves of constant f or x in the P -T plane.

For solar system element composition the aluminium abundance is of the order of 1% of the oxygen abundance. Thus we may neglect with an accuracy sufficient for our purposes compared to . Squaring (50) and equating this to (51) yields

This determines f in terms of x or vice versa. The result can be used in (50) to determine the total pressure P.

We assume x to be held constant. The curve defined in the P -T plane by letting is the limit curve above and to the left of which no spinel exist in an equilibrium state in the presence of corundum. A second curve defined in the P -T plane by letting is the limit curve below and to the right of which no corundum exists in an equilibrium state in the presence of spinel. Between these two limit curves, if they exist ², corundum and spinel coexist in thermodynamical equilibrium. Fig. 12 shows this two limit curves calculated from Eq. (52). The transition from spinel to corundum occurs in a narrow temperature interval of the order of only 1 K, i.e. the transition between both solids is nearly discontinuous in temperature.

Fig. 12. Equilibrium curves for the conversion of corundum (Al2 O3) into spinel (MgAl2 O4). The dashed line shows the upper stability limit of spinel, the full line the lower stability limit of corundum. The dotted line shows the pressure-temperature stratification in the midplane of the accretion disk model.

The limit for conversion of spinel into corundum also is shown in Fig. 9. This limit occurs at a much lower temperature than the limit for conversion of spinel into gas phase molecules. Spinel does not disappear by chemisputtering in a reaction like (47) but by conversion into corundum. This conversion also occurs at a lower temperature than that where corundum is partly converted into gehlenite. Thus, some fraction of the aluminium is bound in gehlenite at the limit where spinel is converted into corundum. The lower stability limit of corundum then is obtained if we let in Eq. (50). This together with Eqs. (44) and (45) then determines the degree g of condensation of calcium in gehlenite and the fraction f of aluminium in corundum resp. in spinel at the border between the region of existence between these two compounds.

5.4. Diopside

The calcium bound in gehlenite (Al₂ Ca₂ SiO₇) at lower temperatures tends to be bound more stable in diopside (CaMgSi₂ O₆). The aluminium liberated in the conversion of gehlenite to diopside does not appear as a gas phase species but forms a solid. Since gehlenite coexists with spinel in the relevant temperature regime, the aluminium content of the gehlenite forms spinel. The magnesium required for this is present as free atoms in the gas phase and the additional silicon and oxygen required to form diopside is present as SiO and H₂ O molecules. A possible reaction for the conversion of gehlenite into diopside then is

In chemical equilibrium the gas phase species satisfy the relation

where . If f denotes the fraction of the calcium bound in diopside and if the only other abundant calcium compound is gehlenite, then the partial pressure of magnesium atoms in the gas phase is since the calcium either is bound in diopside or gehlenite and the aluminium is bound either in gehlenite or spinel. The partial pressures of SiO and H₂ O are and . From the law of mass action (53) we obtain

This defines the curve in the P -T plane along which the fraction f of the calcium is bound in diopside while the remaining fraction of the Ca is bound in gehlenite. At the same time the Al not bound in gehlenite is bound in spinel. defines the upper stability limit of diopside above and to the left of which no diopside exists while defines the lower stability limit of gehlenite below and to the right of which no gehlenite exists ³. These two limit curves are shown in Fig. 13. The transition between the two extreme cases occurs within a very narrow temperature interval of only a few degree where both solids coexist. The transition between the two Ca bearing compounds occurs nearly discontinuous at a sharp transition temperature.

Fig. 13. Equilibrium curves for the conversion of melilite (Al2 Ca2 SiO7) into diopside (CaMgSi2 O6) and spinel (MgAl2 O4). The dashed line shows the upper stability limit of diopside, the full line the lower stability limit of gehlenite. The dotted line shows the pressure-temperature stratification in the midplane of the accretion disk model. The inflection in this line corresponds to the stability limit of iron.

The limit for conversion of gehlenite into diopside also is shown in Fig. 9 from which one easely recognises that the limit occurs at nearly the same temperature but always slightly above that where forsterite starts to be formed with decreasing temperature. This justifies our previous assumption that most of the Mg is present as free atoms in the gas phase where the transition between gehlenite and diopside occurs.

5.5. Anorthite

The aluminium bound in spinel according to calculations of cooling sequences (cf. Grossman 1972, Lattimer et al. 1978) tends to form at lower temperatures the more stable aluminium-calcium compound anorthite (CaAl₂ Si₂ O₈). A conversion of the spinel into anorthite is possible only if the calcium required to form anorthite is taken from diopside which is the only abundant calcium bearing compound. The excess silicon liberated in the destruction of the diopside then will form at the relevant temperatures the magnesium silicate forsterite. A possible reaction for the conversion of spinel and diopside into anorthite and forsterite is

The additional oxygen and silicon atoms required for the conversion are available from H₂ O and SiO molecules from the gas phase. If f denotes the fraction of the aluminium bound in anorthite and g the fraction of the silicon bound in forsterite, the partial pressures of H₂ O and SiO in the gas phase are and since the Ca either is bound in diopside or in anorthite and since the aluminium either is bound in spinel or in anorthite. According to the law of mass action the partial pressures of the gas phase species involved in reaction (55) in chemical equilibrium satisfy the relation

where . It follows from Eq. (56)

On the other hand, the fraction g of the silicon condensed into forsterite is determined by (18) which reads in the present case as

Combining both equations for P yields an equation for f and g which can be solved for g with the result

where

defines the upper stability limit of anorthite while the defines the lower stability limit of spinel where the Al contained in spinel is completely consumed in the formation of anorthite. Fig. 9 shows the result for f. In accord with the finding of Sharp & Huebner (1990) that anorthite does not form in their calculation at temperatures above K the conversion of spinel into anorthite occurs only at rather low temperatures. We do not consider this compound further since it is not present in the inner region of the accretion disk where the silicates and the aluminium compounds are destroyed.

5.6. Aluminium compounds in the disk

According to our above findings the following aluminium compounds are formed in chemical equilibrium in the order of increasing temperature:

1. Anorthite and Diopside. At low temperatures the anorthite consumes all of the available aluminium. Since anorthite contains two Al atoms for each Ca atom and since the Ca abundance exceeds one half of the aluminium abundance, the excess of Ca forms diopside.
2. Spinel and Diopside. As the temperature increases the anortite is converted at a specific temperature into spinel. The spinel then consumes all of the available aluminium. Since spinel contains no Ca, the Ca liberated in the conversion of anorthite into spinel also forms diopside. Thus all of the Ca then is bound in diopside.
3. Spinel and gehlenite. As the temperature increases further the diopside is converted into gehlenite at a specific temperature. The gehlenite then consumes all of the available Ca and a corresponding amount of the aluminium. Only the excess of the Al over Ca remains to be bound in spinel.
4. Gehlenite and corundum. With increasing temperature the spinel is converted into corundum at a specific temperature. At this point of conversion of spinel into corundum, the gehlenite consumes all the available Ca. With increasing temperature, however, the fraction of the Ca bound in gehlenite steadily decreases while the fraction of the Al bound in corundum correspondingly increases.
5. Corundum. At a certain temperature the gehlenite disappears. The corundum then consumes all of the available Al. With continued increase of the temperature the fraction of the Al bound in corundum steadily decreases while the fraction of the Al remainig in the gas phase correspondingly increases. At a specific temperature the last corundum disappears. Beyond that temperature no solid aluminium compunds exist in the disk.

As our discussion of the equilibrium abundances of aluminium compounds shows, this sequence of events occurs in any environment where the pressure and temperature monotonuously increase. Only the precise values of the temperature where the different aluminium 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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