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

4. Thermodynamic properties of the MgO clusters

4.1. Calculation of the free enthalpy

The thermochemical properties of the MgO molecule and of its clusters can be determined once the enthalpy of formation of the clusters and their entropy is known. The enthalpy of formation of a cluster of N molecules from the corresponding free molecules is

where is given in Table (4) and is the heat content of the internal degrees of freedom of the cluster.

The contribution of the vibrational degrees of freedom to the internal energy U of the cluster is

with being the Avogadro number (note that N denotes the number of monomers, which are already diatomic). This is calculated using frequencies calculated in the approximation of small vibrations from eigenvalues of the force-matrix.

The translational, rotational, vibrational and electronic contributions to the partition function of a cluster are assumed to be independent of each other. The entropy in this case is additive in these contributions and we have

for the contribution of the translational degrees of freedom,

for the contribution of the rotational degree of freedom of a linear molecule or

for the contribution of the rotational degrees of freedom for non-linear molecules and

for the vibrational degrees of freedom (see e.g. Benson 1976). M is the mass in atomic mass units and , and are the principal moments of inertia in atomic units (AMU Ų). is the symmetry number of the cluster. The entropy in these equations is in units cal/K.

The contribution of the vibrational degrees of freedom to the entropy S is of the order of R (Benson 1976). This is much smaller than the translational and rotational contributions and, therefore, is neglected. Any contribution of excited electronic states to S can be neglected since such states in small molecules usually have excitation energys above 1 eV and, thus, are not populated at temperatures of the order of 1 000 K or less, which are of interest for circumstellar shells.

From the data given in Table 4 and the equations (34), ..., (38) we can calculate the equilibrium abundance of a cluster of size N from the law of mass action

is the difference between the entropy of one cluster of size N and the entropy of N monomers, the partial pressure of the N -cluster and the partial pressure of the monomers in the gas phase. With the units usually applied in thermochemistry ( in kcal/mol, S in cal/K) and the numerical values in (36), (37) and (38) the pressure is in units of atm.

4.2. The cluster size spectrum

Using the results for the bond energy of the clusters, their moments of inertia the symmetry numbers and the vibrational frequencies we can calculate the abundance of the clusters in a state of thermodynamic equilibrium. The result for a representative H₂ density of cm^-3 typical for the condensation zone of a circumstellar dust shell is shown for four different temperatures in Fig. 4. For the highest temperature K the abundance of the clusters of size relative to the monomer first decreases with increasing cluster size N and, then, increases with increasing cluster size N. This behaviour follows from quite general principles (e.g. Gail and Sedlmayr 1988).

Fig. 4. Density of MgO clusters relative to the monomer density for different temperatures and a gas density of cm-3. The full line connects the most abundant clusters of each size. If this curve decreases for small cluster sizes and increases for large cluster sizes, nucleation occurs. In this case, the least abundant cluster on the curve defines the critical cluster for nucleation

Application of the law of mass action to the heterogeneous equilibrium between the monomer and the solid yields

where is the enthalpy of formation of the solid from the monomers and the corresponding entropy change. is the partial pressure of the monomers in equilibrium with the condensed phase. We define the supersaturation ratio as

and obtain from (40)

is the difference in the bond energies of monomers between the cluster and the solid. This difference is positive since the bond energy of monomers in the solid is higher than in a cluster built from a finite number of monomers. This can clearly be seen from Table 4 or Fig. 1. This bond energy defect results from the reduced attractive interaction energy of monomers with their environment for particles close to the surface as compared to interior particles. In a small cluster, all particles are close to the surface while for the solid such particles form a completely negligible fraction of all particles. In principle, the bond energy defect is roughly proportional to the surface area of the cluster, which in turn for compact structures varies roughly as with the number N of monomers.

For small cluster sizes is a big number since the energy defect for small clusters is of the order of 2 to 3 eV per particle (cf. Fig. 1) while RT for a temperature of K in the condensation zone of a circumstellar dust shell is of the order of eV.

The entropy difference in (43) is dominated by the large and positive contribution of the translational degrees of freedom (cf. Eqs. (36), (37) and (38)), which shows only a weak logarithmic dependence on N. The contributions of the rotational and vibrational degrees of freedom are much smaller and the same holds for the entropy of the solid. The rotational contribution from the cluster shows only a weak logarithmic dependence on N. The entropy term, thus, can roughly be considered as constant.

The term in (43) varies linearly with the cluster size N. It increases for a supersaturated vapour with and it decreases for a subsaturated vapour with . We have to consider three different cases:

- . All three terms in the exponential in Eq. (43) are negative. The equilibrium pressure of clusters relative to monomers decreases with increasing size N, though not necessarily in a monotonic way since there are strong individual scatterings in the bond energy between clusters of comparable size (cf. Fig. 1). The partial pressure of the clusters satisfies the following limit relation

- . The term proportional to increases linearly with increasing N while the term describing the bond energy defect decreases proportional to with increasing N. If

then the exponential in (43) first is dominated for small N by the big negative term proportional to the energy defect and, thus, first decreases with increasing N. For sufficiently large N, however, the exponential becomes dominated by and then increases with N. For intermediate cluster sizes the size distribution has a minimum for a certain value of N. This behaviour can clearly be seen for instance in Fig. 4. Note, however, the strong individual scatterings between neighbouring N.

- and inequality (45) not satisfied. The exponential is always dominated by the term and only increases with increasing N, though not necessarily monotonicly. (The lowest temperature in Fig. 4 is close to this case.) The minimum in the cluster size distribution here is taken for .

In any case, for a supersaturated vapour we have the following asymptotic behaviour of the equilibrium density of clusters for large N:

© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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