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

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3. Annealing and diffusion I

It is generally believed that the internal microstructure of the lattice of interstellar and molecular cloud dust particles to a large extent is amorphous (cf. the review by Dorschner & Henning 1995). The dust within the protoplanetary disk is of precisely such an origin. It has fallen during the collapse phase from the molecular cloud onto the accretion disk. During the viscous stage of the protoplanetary disk the central protosun has already grown to nearly its final mass by accreting most of the mass contained in the disk. The remaining material within the accretion disk in this phase of its evolution essentially consists of material from the parent molecular cloud which most likely has landed on the disk at rather large distances ([FORMULA] AU) from the protostar (Cassen 1994). The infalling cloud material prior to its mixing with previously accreted disk matter passes through an accretion shock standing on both sides of the disk surface. In the outer parts of the accretion disk the strength of this shock is not particularly high and the dust component is not strongly heated or even destroyed during its passage through the shock (e.g. Lunine et al. 1991, Neufeld & Hollenbach 1994, Ruzmaikina & Ip 1994). Only ice coatings on the grains are vapourised, but in the outer parts of the disk at distances of [FORMULA] AU and more they are formed anewed when the grains and the gas cool behind the shock. The dust component of the disk matter during the viscous stage of disk evolution, thus, is expected to be almost unprocessed dust from the parent molecular cloud.

The inwards transport of matter during the viscous evolution of the disk carries the dust particles slowly into the hot central parts of the accretion disk where the dust ultimately is destroyed either by thermal decomposition or by chemical surface reactions at temperatures above 1 000 K for the carbonaceous dust component (Finocchi et al. 1997) and above 1 500 K for the silicate dust component (e.g. Duschl et al. 1996). At a somewhat lower temperature the dust already is subject to the process of annealing (Lenzuni et al. 1995, Duschl et al. 1996). At elevated temperature the lattice vibrations in the dust become sufficiently excited that activation energy barriers can be overcome such that atoms or groups of atoms may change their position or their orientation within the lattice. The atoms in the amorphous dust material then start to rearrange and to migrate into energetically more favourable positions or orientations within the lattice, where they are more tightly bound and, then, become less mobile. By this process the dust material gradually develops some kind of local order and slowly changes its lattice structure from the strongly disordered structure of an amorphous material to the locally ordered structure of a microcrystalline material. If the amorphous dust grains contain a significant fraction of impurity elements within their lattice, for instance Al and Ca replacing some of the kations or anions in the olivine lattice, then the annealing of lattice defects is accompanied by chemical fractionation within the grain. The impurities are gathered in separate inclusions or migrate to the grains surface. If annealing lasts sufficiently long even complete crystallisation of the previously amorphous grains may occur.

3.1. Silicate grains

The basic microscopic processes responsible for annealing are diffusion of vacancies and interstitials, and self-diffusion. For the purpose of a rough estimation these processes can be approximated by a 3D random walk on a cubic lattice. The diffusion coefficient in this case is

[EQUATION]

(e.g. Dekker 1963). a is the average step length, [FORMULA] is the number of attempts per unit time for hopping to a neighbouring lattice site and [FORMULA] is the activation energy. The characteristic activation energy for silicate materials has been estimated by Lenzuni et al. (1995) and Duschl et al. (1996) to be [FORMULA] K based on the annealing experiment of Nuth & Donn (1982) for condensates from magnesium-silicate smokes and assuming the characteristic frequency [FORMULA] to equal the average vibrational frequency [FORMULA] s-1 of the SiO4 tetrahedron. Essentially the same [FORMULA] results if one uses the Debye temperature [FORMULA] K for Mg2 SiO4 (Kuskov & Galimzyanov 1986) for the determination of the characteristic frequency [FORMULA]. The characteristic length a is estimated from the volume V of the basic molecule forming the lattice:

[EQUATION]

[FORMULA] is the molecular weight of Mg2 SiO4, m the atomic mass unit and [FORMULA] g/ cm3 the mass density of Mg2 SiO4. We obtain for the coefficient of solid state diffusion within the silicate

[EQUATION]

In a 3D random walk the average rms displacement [FORMULA] within time t is given by

[EQUATION]

Fig. 1 shows the typical time t required for a single atom to walk over some prescribed distance d. This may be identified with the time required for annealing the amorphous structure of unprocessed grains from the parent molecular cloud and to form at least a local crystal structure extending over regions of size d. We arbitrarily assume that diffusion over a distance of [FORMULA] m is required to convert the amorphous dust material into a locally ordered structure with (poly)crystalline properties. The time required for this has to be compared with the duration of the viscous stage of disk evolution which lasts for roughly [FORMULA] yrs. From Fig. 1 we infer that at temperatures above [FORMULA] K in the midplane of the disk the grains loose their amorphous structure by local rearrangement and gain an ordered lattice structure. We also can compare the diffusion timescale with the characteristic timescale [FORMULA] for changes of the temperature in the frame of a gas parcel drifting towards the star, which is shown as a dashed line in Fig. 1. This also yields a limit temperature of [FORMULA] K above which the amorphous silicate dust material is converted into a crystalline one. The same conclusion has been arrived at in Duschl et al. (1996) by a slightly different argument. A laboratory experiment with striated orthopyroxene showed annealing of the structure by heating for one week to 1 100 K (Ashworth et al. 1984), roughly consistent with our estimation of diffusion timescales.

[FIGURE] Fig. 1. Time scale t (in s) for solid state diffusion at temperature T within silicate dust particles of the indicated size (in µm). The dashed line shows the characteristic timescale [FORMULA] for temperature changes in the accretion disk given by Eq. (68).

This annealing of any amorphous structure of the grains inherited from their circumstellar birth conditions allows to apply thermochemical data measured in the laboratory for crystalline materials to dust materials in protoplanetary disks, if one calculates dust compositions, vapourisation temperatures etc. It also requires to use data for crystalline dust materials in calculating the opacity in regions where the dust is heated to temperatures [FORMULA] K. This is discussed in detail within the context of model calculations in Sect.  7.

3.2. Iron grains

Besides silicate grains, iron metal grains are likely to exist in the accretion disk. For iron the measured value (by radioactive tracer diffusion) of the coefficient of self-diffusion is

[EQUATION]

(Weast 1981). The diffusion timescale for iron grains at a temperature of 1 000 K is approximately [FORMULA] -times shorter than the diffusion timescale for the silicates. Any possible amorphous structure of the iron, therefore, is removed already at rather low temperatures. An inspection of Fig. 2 shows that at temperatures above [FORMULA] K iron grains should develop at least locally a well ordered structure and above [FORMULA] K even the biggest grains of interstellar origin had enough time to rearrange into a well ordered lattice structure extending over the whole grain.

[FIGURE] Fig. 2. Time scale t (in s) for solid state diffusion at temperature T of sulphur atoms within iron particles of the indicated size (in µm). The vertical line shows the limit temperature for the conversion of Fe into FeS in the disk model. The dashed line shows the characteristic timescale [FORMULA] for temperature changes in the accretion disk given by Eq.  (68).

Many other abundant possible impurity atoms have diffusion coefficients with a very similar activation energy and frequency factor, for instance the metals Ni and Mn or the non-metals S and P (Weast 1981). They can easely move around within the lattice which means that only components which are easely soluble in the iron metal remain inside the iron grains (Ni for instance, which then forms separate Ni-Fe crystals; this process has been studied in a computer simulation by Willis & Goldstein 1981) while non soluble elements most likely are driven out of the lattice and assemble at the surface of the grains from which they are lost to the gas phase. One can expect, thus, that iron grains in the warm part of an accretion disk are rather clean metal clusters.

The ease with which certain atoms may diffuse through the iron during the long period of time available as the grains slowly migrate inwards is especially important for the sulphur since at low temperature part of the iron forms FeS (see Sect.  4.3). The diffusion coefficient of S atoms in iron metal is

[EQUATION]

(Weast 1981). The resulting characteristic diffusion timescale is shown in Fig. 2. The timescale at the stability limit of FeS is short enough even for micron sized grains that sulphur atoms may diffuse into or out of iron grains in order that the conversion of FeS to Fe or vice versa is possible.

3.3. Carbonaceous grains

Annealing of an amorphous structure is not possible for the carbonaceous dust component since this requires breaking of the strong C-C-bonds. The coefficient of self-diffusion for C in carbon is

[EQUATION]

(Weast 1981). Diffusion time scales at 1 000 K then are roughly 15 orders of magnitude longer than for silicates. Annealing of interlayer bonds from partially ordered carbon, however, is possible at medium temperatures. After processing a carbonaceous material to a temperature [FORMULA] K such bonds are essentially removed (Rietmeijer & Mackinnon 1985).

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© European Southern Observatory (ESO) 1998

Online publication: March 30, 1998
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