3 Dust growth and evaporation

A fundamental process for the consideration of the time-dependent
behaviour of the dust component in brown dwarf atmospheres is the
growth of the dust particles by accretion of molecules. The
respective reverse process (thermal evaporation, in view of more
complex surface reactions also sometimes denoted by chemical
sputtering) is important at high temperatures^{}. Considering the thermodynamical conditions
in brown dwarf atmospheres, we are again faced with the problem of
qualitative changes of the dynamical behaviour of the gas component
due to different Knudsen numbers.

For large Knudsen numbers (*lKn*), gas molecules of all kinds are freely
impinging onto the surface of the grain. Some of these dust-molecule
collisions (sometimes a certain sequence of them) will initiate a chemical
surface reaction which causes a growth step (or an evaporation step) of the
dust particle. This case has been extensively studied in the circumstellar
envelopes of AGB stars (Gail
Sedlmayr 1988; Gauger
1990;
Dominik
1993).

The accretion rate, expressed in terms of the increase of the
particle's volume
due to chemical surface
reactions for large Knudsen numbers is given by

where "r'' is a general surface reaction index, is the increase of the dust particle's volume

The last term on the r.h.s. of Eq. (22) takes into
account the reverse chemical processes, namely the thermal evaporation
rates. It determines the sign of
and hence decides whether
the dust particle grows or shrinks.
is
a generalised supersaturation ratio of the surface reaction "r''
(Dominik
1993) where
is the particle density of the key
species in phase-equilibrium over the condensed dust material.
is not known a priori. In the case of simple surface
reactions, which transform
units of the solid material from the gaseous into the condensed phase (and vice versa), e.g.
(
,
)
or
(
,
), the generalised
supersaturation ratio
is related to the usual supersaturation ratio
of the dust grain
material
(Gauger
1990) by

(23) |

denotes the saturation vapour pressure of the molecule over a flat surface of the condensed state, which is a pure temperature function determined by the Gibbs free energies of the solid and the gaseous species . The supersaturation ratio

For small Knudsen numbers (s*Kn*), the transport of gas molecules to the
surface of the grain (or the transport of evaporating molecules away
from the grain's surface) is not a simple free flight with thermal
velocity as assumed in Eq. (22), but is hindered by
inter-molecular collisions. Consequently, the grain growth and
evaporation is limited by the diffusion of molecules towards or away
from the grain's surface, considering growth or evaporation in the
laminar case, respectively (we disregard here convection as transport
process for the molecules, expected to occur in the turbulent case).
We consider the following particle conservation equation with a
diffusive transport term (see Landau & Lifschitz 1987, Eq. (38.2)
ff)

is the concentration of gaseous molecules of kind

where

is the thermal velocity of a gas particle of kind

Considering the static case (
)
and assuming stationary
(
)
and spherical symmetry,
Eq. (24) results in

Assuming furthermore constant and

In order to solve the second order diffusion equation (27), two boundary conditions must be specified. First, considering the asymptotic behaviour for , we assume that the concentration

i.e.we assume that the chemical surface reactions responsible for the growth and evaporation of the dust particle are sufficiently effective to completely exhaust or enrich the gas in the boundary layer with molecules of kind

The solution of Eq. (28) with the boundary
conditions (29) and (30) is

and the the total volume accretion rate of the grain for small Knudsen numbers, summing up the contributions of several surface reactions of index r (like in Eq. (22)) with net rate , according to Eqs. (25) and (31), is

The description of the volume accretion rate according to Eq. (32) again allows for the simultaneous growth of different solid materials on the same surface, resulting in "dirty'' grains (heterogeneous growth).

For arbitrary Knudsen numbers we apply the same interpolation scheme
as outlined in Sect. 2.3. We define a critical Knudsen
number
by equating Eqs. (22) with (32)

(33) |

Using Eqs. (26) and (11), the result is

(34) |

Assuming perfect sticking ( ) and considering typical molecular radii and masses between Å and Å, and between amu and amu, respectively, the resulting critical Knudsen numbers for growth are found to range in 0.15 to 0.24, independent of density and temperature. Thus, we simply adopt an unique critical Knudsen number for all growth and evaporation species

(35) |

Our ansatz of the general volume accretion rate for arbitrary Knudsen numbers with is

Figure 3 shows the resulting growth time-scale

(37) |

as function of particle size

where SiO is identified as the key educt.

The particle growth is found to be typically 3 orders of magnitude slower than the acceleration which allows us to assume instantaneous acceleration (equilibrium drift). For large Knudsen numbers, we find whereas for small Knudsen numbers the growth time-scale increases faster for larger grains and becomes density-independent, . Note that the influence of the drift velocities on the particle growth has not been considered in Eqs. (22) and (32) such that for supersonic drift velocities or large dust Reynolds numbers, the presented physical description is not valid.

3.1 Maximum grain size

An additional dashed line is depicted in Fig. 3, where . This line defines a maximum dust grain size in a brown dwarf atmosphere. For larger particles ( above the dashed line), the growth time-scale exceeds the time-scale for gravitational settling ( ) which means that such particles are already removed from the atmosphere before they can be formed. Consequently, such particles cannot exist. The maximum grain size varies between 1 m in the thin, outer atmospheric regions ( ) and 100 m in the dense, inner regions ( ). These values depend on the stellar parameters and the considered dust material density.

Note, that an absolute minimum of has been considered in Fig. 3 since extreme supersaturation ( ) and solar abundance of silicon in the gas phase have been assumed. In the case of an Si-depleted or nearly saturated gas (), becomes larger and the maximum particle radius becomes smaller. Furthermore, the values for are relatively independent of temperature, but will shift as in the free molecular flow case and in the laminar viscous case, remembering that in both cases and that (see also Eq. (81)).

Figure 3 demonstrates furthermore that dust particles
moving with supersonic drift velocities cannot be expected in brown
dwarf atmospheres. Similarly, the turbulent flow regime with dust
Reynolds numbers
is barely reached at
very large densities. Therefore, we can conclude that for dust grains in
brown dwarf atmospheres the *subsonic free molecular flow* and the
*laminar viscous flow* are the important cases to be investigated.

Copyright ESO 2003