## 5. Application to nucleation in stellar winds## 5.1. Calculation of dust nucleationThe theoretical aspects of the dust formation and destruction processes are discussed in detail by Gail and Sedlmayr (1987). We restrict our considerations in this paper to the formation of MgO clusters and homogeneous particle growth, and neglect chemical sputtering and heteromolecular growth processes. In the general case of dust formation more than one growth species, like dimers and trimers, could be involved in the growth process. As we shall see, in our case the monomer can be adopted as the only growth species for MgO dust formation. Then the rate equations describing the nucleation process become much simpler. Our derivation of the nucleation rate given below is a simplified version adapted to the case of MgO nucleation of the more general discussion given in Gail and Sedlmayr (1988). The net rate of change of the number density
of clusters of size where , are the gain
and loss rates from clusters of size to
clusters of size where is the root mean square thermal velocity of the monomers impinging onto the surface of an is the density of We define an effective transition rate In terms of the effective transition rates the master equations (47) can be written as Next we define the total density of dust grains by From (55) we obtain by summing the equations with since there is no contribution from terms with
since arbitrarily big clusters cannot be
formed. is the with which denote the growth and evaporation rate, respectively. (58) is
a system of equations for and
in a quasi-stationary state. It holds for all
We draw two conclusions from this: - : As is discussed in sec. 4.2, for the size distribution satisfies the limit relation (44). The physically plausible solution in this case is Then, in the stationary case, the actual size distribution equals the size distribution in the thermodynamic equilibrium state. - : For sufficiently large Since there cannot exist arbitrarily large clusters,
has to vanish above some (possibly very large)
and the second term on the r.h.s. of (61) then
goes to zero for sufficiently large Usually, for has a
sharp minimum in the sense that the smallest value of
taken at some is much
smaller than the value of for any other
Thus we have the very plausible result that the rate of dust particle formation is determined by the slowest growth step on the reaction chain along clusters of increasing size which occurs at the cluster size where is smallest. (Note, however, that there exists a different possibility: If the
least stable cluster with size Using (64) in (61) we obtain for the size distribution in the stationary case For both sums can be approximated by the term with . This shows i.e. in the subcritical region the size distribution equals the thermodynamic equilibrium size distribution. In this cluster size regime the relaxation time of the size distribution towards equilibrium is shorter than the slow changes of the total number of clusters with size introduced by the leakage at from subcritical clusters to the realm of dust grains with . For the sum in the denominator can be
replaced by the term with . In the nominator,
the equilibrium size distribution is
increasing with increasing This is the size distribution in a stationary equilibrium state which, however, is not realized for arbitrarily large cluster sizes (see Gail and Sedlmayr 1988). ## 5.2. Application to MgO dust formation in stellar windsWe are now prepared to determine the nucleation rate of MgO in
circumstellar shells of M-stars. Fig. 5 shows for a typical
hydrogen density cm
The particle densities of the critical cluster are extremely small
and make nucleation of MgO in circumstellar dust shells around M-stars
as a dust species of its own or as seed nuclei for a different dust
component completely impossible. At 850 K for instance we have a
typical cluster density of the critical cluster of
cm This does not result from a principal inability of MgO to form clusters but from the low bond energy of the MgO molecule, which according to recent determinations is only 3.5 eV (and, perhaps may be even lower, cf. Sect. 2.4). The high abundance of hydrogen relative to both magnesium and oxygen on the one hand and the somewhat higher bond energy of 4.4 eV of the OH bond on the other hand makes it much more favorable for the oxygen atom to form an OH bond instead of a bond with Mg. Magnesium remains in this case in the gas phase mainly as the free Mg atom. The situation would be quite different if the bond energy of MgO were only slightly higher, as older determinations of the dissociation energy of MgO ( eV) suggested, since then the formation of MgO would become more favourable and the resulting much higher gas phase abundance of MgO would result in efficient MgO nucleation (Gail and Sedlmayr 1986). Unfortunately, the precise nucleation rate depends critically on the uncertain value of the bond energy of MgO and a more definite conclusion with respect to the possibility or impossibility of MgO nucleation can only be drawn when a definitive value for the bond energy becomes available. Also, a less H-rich environment would increase the MgO abundance and MgO nucleation could then become an efficient process. Finally it should be noted that the present calculation assumes equal numbers of Mg and O atoms in a cluster. In principle it is possible that non-stoichiometric clusters form by adding the more abundant Mg atoms to a cluster. Such clusters may exist in nature and addition of the abundant Mg and reactions with OH may significantly increase the growth rate and abundance of MgO clusters, but such clusters cannot be treated by the present potential model. © European Southern Observatory (ESO) 1997 Online publication: June 30, 1998 |