Astron. Astrophys. 320, 553-567 (1997)

5. Application to nucleation in stellar winds

5.1. Calculation of dust nucleation

The 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 N is the sum over the rates of all possible gain and loss processes of the N -cluster and is described by the master equation for the time evolution of

where , are the gain and loss rates from clusters of size to clusters of size N and vice-versa, and , the corresponding gain and loss rates from to N. These rates are given by

where is the root mean square thermal velocity

of the monomers impinging onto the surface of an N -cluster, the surface area of the N -cluster and the sticking coefficient for monomers in collisions with an N -cluster. is the vapourization rate per unit area of the surface. This rate can be determined from the principle of detailed balance between the growth process and its inverse process in a thermodynamic equilibrium state. In this state the growth and evaporation processes are in mutual equilibrium which requires

is the density of N -clusters in equilibrium with the monomers in a thermodynamic equilibrium state. Eq. (53) defines in terms of and equilibrium densities. This Milne relation for can be applied also in non-TE situations provided that the internal states of the N -cluster are populated thermally with some excitation temperature , which in circumstellar shells is likely to be different from the gas kinetic temperature (e.g. Nuth and Donn 1981). We neglect here any non-TE level-population effects.

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 nucleation rate which we call . As has been shown in Gail and Sedlmayr (1988) in a stellar wind with a sufficient mass-loss rate the densities relax to a quasi-stationary equilibrium state for all N up to a certain . The effective transition rates then are independent of N and equal to . This means

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 N with . It can be solved by eliminating successively , , , ... between consecutive equations (58) for , ..., . The result is (Gail and Sedlmayr 1988)

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 N, is given by (46). This means

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 N. The nucleation rate , then, is given by

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 N. This can clearly be seen, for instance, in Fig. 4. The sum in (64) then is determined by its biggest term ². The contribution of all other clusters of size can usually be neglected. Then the nucleation rate is given by

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 N has an extremely low abundance compared to the cluster with size it is possible that the growth step from to by dimer addition becomes more efficient than the growth step from N to by monomer addition. Than the step from to becomes the rate determining step.)

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 i for and for sufficient (cf. Eq. (46)) and small N the sum can be replaced by the single term with . Then

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 winds

We 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^-3 within the condensation zone of a circumstellar dust shell the cluster densities of clusters in the temperature range between 500 and 1500 K where circumstellar condensation is likely to occur. The partial pressure of MgO in the gas phase is calculated from chemical equilibrium in the gas phase considering the most abundant molecular species and the abundance of the clusters is calculated from (40). The critical cluster for nucleation according to our results obtained in the preceding section is that one for which the equilibrium density is smallest for a given density and temperature. The size of the critical cluster and its particle density then can be determined from the lower envelope of the family of curves which is shown in Fig. 5. We read off from the figure that the critcal cluster size above K is , between and 850 K is and below K the critical cluster is the monomer itself. The nucleation rate then can easily be calculated from Eq. (65).

Fig. 5. Clusters in chemical equilibrium with the monomer at a hydrogen density of cm-3 which is typical for the condensation zone. Shown are the densities for clusters of size , ...,

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^-3 and a typical collision frequency of the critical cluster with MgO molecules of s^-1. With a typical width of the condensation zone at the inner edge of the dust shell of cm and an expansion velocity of cm s^-1 near the sonic point the gas typically requires one year to cross the condensation zone of a circumstellar dust shell. Then we would obtain roughly dust grains per hydrogen molecule. A lower condensation temperature of 650 K will increase this number to only grains per hydrogen molecule while real dust shells contain roughly dust grains per hydrogen nucleus. MgO nucleation, thus, is completely negligible, even as a subordinate dust component.

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
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