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

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7. Annealing and diffusion II

The silicate dust grains exist in the accretion disk in at least three different modifications: (i) As the grains enter the accretion disk from the parent molecular cloud they have most likely an amorphous lattice structure. The P94 model assumes a mixture of silicate compounds as specified in Tab. 1. (ii) The components of this mixture are thermochemically unstable in the warm inner parts of the accretion disk. The most stable condensed silicate compounds in an environment with solar system composition are enstatite with a very low iron content and (iii) above the stability limit of the enstatite grains, an iron-poor olivine with nearly the composition of forsterite is the thermodynamically most stable silicate compound. Thus, if the disk material drifts inwards we expect that with increasing temperature first internal annealing processes convert the grains from an amorphous to a crystalline structure (see Sect.  3). Impurities which do not solve in the silicate crystal assemble most likely in localised inclusions within the polycrystalline material and may crystallise into small particulates of a different chemical composition within the bigger silicate grains. This especially is expected to occur for the iron of which only as small fraction is incorporated into the orthopyroxene and olivine grains in chemical equilibrium at temperatures above 800 K (Saxena & Ericksson 1986). If the internal structure changes to that of enstatite the excess of the iron content of silicates in the P94 mixture over the equilibrium concentration most likely assembles in small iron inclusions within the polycrystalline material. Also complete removing of the Fe by diffusion to the surface and loss to the gas phase may occur at sufficiently high temperature during complete crystallisation of the grain.

At a temperature of roughly 1 200 K the conversion of enstatite to forsterite becomes thermodynamically favourable (cf. Fig. 8). Finally the forsterite grains are destroyed when the disk matter crosses the stability limit of forsterite. Any silicate grain, thus, suffer the following sequence of transmutations on its outside-in journey in the disk:

[EQUATION]

In the annealing experiment of Nuth & Donn (1982) it was found that forsterite forms from an amorphous silicate smoke by annealing for up to 30 hours at 1 000 K, but not enstatite, though the enstatite is the thermodynamically favoured compound. Thus, possibly the forsterite grain component in the initial P94 mixture cannot easely be converted into enstatite by solid state annealing and retains its initial composition until it is destroyed in the inner part of the disk. Also the enstatite grain component in the P94 mixture may be not easely converted into forsterite in the region where forsterite is thermodynamically more stable than enstatite and survives until it is destroyed. In this paper we shall assume that the conversion occurs with a sufficiently small timescale compared to the characteristic timescale of [FORMULA] yrs for temperature changes such that the relative fractions of forsterite and enstatite always correspond to the chemical equilibrium state.

With respect to the disk structure the most important of these changes of dust properties is the transition from amorphous to polycrystalline dust since this is accompanied by a considerable change in the absorption properties of the dominant absorber, as can be seen from Fig. 14. In order to account for this process in a model calculation for the disk structure we consider the following crude model for the annealing process by internal hopping and re-arrangement processes: We assume that there exist certain tiny islands of a few geometrically well ordered SiO4 tetrahedrons within the random network of SiO4 groups of the amorphous dust, which serve as "growth centres" for the formation of an ordered lattice structure. Once some SiO4 tetrahedron from the random network adjacent to such a growth center has re-arranged its position and orientation relative to the growth center by internal hopping such that it now fits well to the structure of the growth centre it finds itself in a deeper local minimum of the lattice potential than in its previous position (orientation). It is, then, much more likely that the next adjacent tetrahedron aligns to the growth center as that the former SiO4 group gives up its energetically favourable state. The growth centre then has increased in size by one building block of the lattice. We then assume that the conversion of the amorphous dust material into a polycrystalline material occurs by growth of the well ordered region into its disordered environment, starting at some tiny well ordered centres which always exist in a disordered material.

Let [FORMULA] be the volume occupied by a single building block of the lattice and let V be the actual volume of a growth centre. The increase of V per unit time is given by

[EQUATION]

[FORMULA] is the number of surface sites of the growth centre (which we take for simplicity to be a cube) where a new building block can be added from the environment. [FORMULA] is the frequency of attempts of particles from the environment to jump into an energetically more favourable state by aligning to the surface of the growth centre and [FORMULA] is the activation energy barrier to be surmounted in a successful transition. For [FORMULA] and [FORMULA] we use the values given in Sect.  3for annealing of amorphous silicates.

Eq. (67) integrates to

[EQUATION]

Since the integrand is steeply increasing near the upper limit of integration, we can expand the exponent around [FORMULA]

[EQUATION]

and then integrate with the result

[EQUATION]

In Fig. 15 we show the number [FORMULA] of building blocks contained in one growth centre as a function of the temperature in the central plane of the disk. [FORMULA] and [FORMULA] are calculated in this case from the equations for the disk structure with the simple assumption of [FORMULA] cm2 / g. If we arbitrarily assume that we have initially one growth center per [FORMULA] Si atoms in the dust material we find a temperature of [FORMULA] K for the transition from an amorphous to a polycrystalline structure. This agrees with our previous estimate in Sect.  3since a [FORMULA] silicate grain contains roughly [FORMULA] Si atoms. Due to the steep increase of V with T our estimate of the transition temperature is rather robust and depends only weakly on the precise number of growth centers which are really present in the dust material.

[FIGURE] Fig. 15. Number N of building blocks of the silicate lattice contained in one growth centre if the dust temperature has increased to the temperature T.

As a crude measure for the degree of conversion of the amorphous into polycrystalline grains we introduce the ratio of the total volume of all growth centres to the total volume of all silicate grains

[EQUATION]

We do not discriminate between the different silicate dust components in the P94 mixture. The total volume of all centres is [FORMULA] where x the concentration of growth centres per Si nuclei which we choose to be [FORMULA] and [FORMULA] is the density of hydrogen nuclei. The total volume of all silicates is given by [FORMULA]. It follows from (69) and the disk equations

[EQUATION]

If the calculated value of [FORMULA] exceeds unity we have to put [FORMULA].

The pressure at the transition point is approximately 50 dyn/ cm2. In chemical equilibrium at the transition point at this pressure and a temperature of 800 K the orthopyroxene and olivine both are nearly iron free (cf. Figs.  20 and 21 in Saxena & Ericksson 1986 for instance). The transition temperature is well above the temperature for formation of FeS, i.e., the iron is present as the free metal. After annealing the iron content of the silicates in the initial P94 mixture then forms separate iron particulates. Whether they form inclusions inside the silicate grains or whether the excess iron outgasses and precipitates on the iron grains contained in the P94 mixture is an open question which can only be answered by a detailed study of the underlying transport processes which is out of the scope of the present paper.

The steep increase of [FORMULA] with T means that the transition from amorphous to polycrystalline grains occurs within a rather narrow temperature interval. This transition has strong implications for the disk structure since during this transition the mass extinction coefficient drops by nearly two decades, as the extinction changes from that of the amorphous silicate to that of a mixture of olivine and iron particles (see Fig. 14). In view of the uncertainties as whether part of the iron partially is included as small inclusions in the silicates or not we calculate the extinction coefficient in the transition region as a simple linear superposition of the contributions of the three main absorbers

[EQUATION]

The contribution of iron to [FORMULA] considers that 49% of the iron in the P94 mixture is already present as iron grains in the region of the disk where annealing occurs while 51% is bound in the silicates prior to annealing (see Table 1) and has precipitated onto iron grains thereafter.

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© European Southern Observatory (ESO) 1998

Online publication: March 30, 1998
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