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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.

Free molecular flow ( ${Kn}\!\gg\!1$):

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 ${\rm\hspace*{0.7ex}\&\hspace*{0.7ex}}$Sedlmayr 1988; Gauger ${\rm\hspace*{0.8ex}et\hspace*{0.7ex}al.\hspace*{0.5ex}}$1990; Dominik ${\rm\hspace*{0.8ex}et\hspace*{0.7ex}al.\hspace*{0.5ex}}$1993).

The accretion rate, expressed in terms of the increase of the particle's volume $V\!=\!4\pi a^3/3$ due to chemical surface reactions for large Knudsen numbers is given by

 \begin{displaymath}{\frac{{\rm d}V}{{\rm d}t}}^{\rm lKn} =
4\pi~a^2 \sum\limit...
..._{\rm r}
\alpha_{\rm r} \left(1-\frac{1}{S_{\rm r}}\right) ,
\end{displaymath} (22)

where "r'' is a general surface reaction index, $\Delta V_{\rm r}$ is the increase of the dust particle's volume V caused by one reaction "r''. $n_{\rm r}$is the particle density of a key gas species whose collision rate limits the rate of the surface reactions of index "r''. This gas species has to be identified by considering the particle densities present in the ambient gas and stoichiometric constraints. Equation (22) explicitly allows for the consideration of heterogeneous growth, where different solid phases can grow simultaneously on the same surface, resulting in "dirty'' grains. The relative velocity is here defined as $v^{\rm rel}_{\rm
r}\!=\!\sqrt{kT/(2\pi m_{\rm r})}$ where $m_{\rm r}$ is the mass of the key species. $\alpha_{\rm r}$ is a sticking coefficient which contains more detailed knowledge about the surface chemical process, if available.

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 ${\rm d}V/{\rm d}t$ and hence decides whether the dust particle grows or shrinks. $S_{\rm r}\!=\!n_{\rm r}/n_{\rm r}^{\hspace{-0.9ex}^{\circ}}$ is a generalised supersaturation ratio of the surface reaction "r'' (Dominik ${\rm\hspace*{0.8ex}et\hspace*{0.7ex}al.\hspace*{0.5ex}}$1993) where $n_{\rm r}^{\hspace{-0.9ex}^{\circ}}$ is the particle density of the key species in phase-equilibrium over the condensed dust material. $S_{\rm r}$ is not known a priori. In the case of simple surface reactions, which transform $m_{\rm r}$ units of the solid material $\cal{A}$from the gaseous into the condensed phase (and vice versa), e.g. $~\rm
[TiO_2]_N(s) + TiO_2 \rightleftharpoons [TiO_2]_{N+1}(s)$ ( $\cal{A}\!=\!\rm
TiO_2$, $m_{\rm r}\!=\!1$) or $\rm C_N(s) + C_2H_2 \rightleftharpoons
C_{N+2}(s) + H_2$ ( $\cal{A}\!=\!\rm C$, $m_{\rm r}\!=\!2$), the generalised supersaturation ratio $S_{\rm r}$ is related to the usual supersaturation ratio $S\!=\!n_{\cal{A}} kT/p_{\cal{A}}^{\rm vap}(T)$ of the dust grain material $\cal{A}$ (Gauger ${\rm\hspace*{0.8ex}et\hspace*{0.7ex}al.\hspace*{0.5ex}}$1990) by

\begin{displaymath}S_{\rm r} = S^{m_{\rm r}} .
\end{displaymath} (23)

$p_{\cal{A}}^{\rm vap}\!=\!n_{\cal A}^{\hspace{-0.9ex}^{\circ}}kT$ denotes the saturation vapour pressure of the molecule $\cal{A}$ 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 $\cal{A}$. The supersaturation ratio S is well-defined even if $\cal{A}$ is not stable as a free molecule (Woitke 1999). It is principally possible to account for additional effects caused by a fast relative motion of the grain within the scope of this description for grain growth and evaporation. These effects comprise an enlargement of $v^{\rm rel}_{\rm r}$ as well as a decrease of $\alpha_{\rm r}$ in the case of super-thermal collisions (see Krüger et al. 1996). However, such effects become only relevant for supersonic drift velocities which, following the discussion in Sect. 2.4, are not very likely to be relevant for brown dwarf atmospheres due to the fast self-removal of the respective dust particles.

Viscous case ( ${Kn}\!\ll\!1$):

For small Knudsen numbers (sKn), 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)

 \begin{displaymath}\frac{{\rm\partial} }{{\rm\partial} t}(\rho~c_i) + \nabla~(\vec{v}_{\rm gas}~\rho~c_i) = - \nabla \vec{j}^{\rm diff}_i.
\end{displaymath} (24)

$c_i\!=\!n_i/\rho\;\rm [g^{-1}]$ is the concentration of gaseous molecules of kind i in the gas with mass density $\rho $ and velocity $\vec{v}_{\rm gas}$. The diffusive particle flux is given by

 \begin{displaymath}\vec{j}^{\rm diff}_i= - \rho~D_i\nabla c_i ,
\end{displaymath} (25)

where Di is the diffusion constant of gas particles of the kind i in the gas mainly composed of H2 molecules (we neglect He atoms here), as given by Jeans (1967, Eq. (260))

 \begin{displaymath}D_i = \frac{v_{{\rm th},i}^{\rm red}}{3 \pi~(r_{\rm H_2}\!+\!r_i)^2~ n}\cdot
\end{displaymath} (26)

$v_{{\rm th},i}^{\rm red}\!=\!\sqrt{8kT/(\pi m_{\rm red})}$ is the thermal velocity of a gas particle of kind i with reduced mass $1/m_{\rm red}\!=\!1/m_{\rm H_2}\!+\!1/m_i$, where $m_{\rm H_2}$ and mi are the masses of H2 and i, respectively. $n\!=\!\rho/\bar{\mu}$ is the total gas particle density. The particle radii ri can be derived from friction experiments (see Jeans 1967, p. 183) and vary between the values for hydrogen $r_{\rm
H_2}\!=\!1.36$ Å, carbon monoxide $r_{\rm CO}\!=\!1.89$ Å, water and carbon dioxide $r_{\rm H_2O}\!=\!r_{\rm
CO_2}\!=\!2.33$ Å to benzene $r_{\rm C_6H_6}\!=\!3.75$ Å.

Considering the static case ( $\vec{v}_{\rm gas}\!=\!0$) and assuming stationary ( $\partial (\rho~c_i)/\partial t\!=\!0$) and spherical symmetry, Eq. (24) results in

 \begin{displaymath}\nabla\big(\rho~D_i\nabla c_i) =
\frac{1}{r^2} \frac{{\rm\p...
...2\rho~D_i\frac{{\rm\partial} c_i}{{\rm\partial} r}\bigg) = 0 .
\end{displaymath} (27)

Assuming furthermore constant $\rho $ and Di, we find

 \begin{displaymath}r^2\rho~D_i\frac{{\rm\partial} c_i}{{\rm\partial} r} = {\rm const}_i .
\end{displaymath} (28)

In order to solve the second order diffusion equation (27), two boundary conditions must be specified. First, considering the asymptotic behaviour for $r\!\to\!\infty$, we assume that the concentration ciapproaches the undisturbed value $n_i/\rho$ in the distant gas. Second, at the lower integration boundary $r\!=\!a$, we assume phase equilibrium
         $\displaystyle c_i(\infty)$ = $\displaystyle \frac{n_i}{\rho}$ (29)
ci(a) = $\displaystyle \frac{n_{i}^{\hspace{-0.9ex}^{\circ}}}{\rho} = \frac{n_i}{\rho}\frac{1}{S_i} ,$ (30)

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 i for growth and evaporation, respectively. As in the last paragraph, $S_i\!=\!n_i/n_{i}^{\hspace{-0.9ex}^{\circ}}$ is the supersaturation ratio.

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

 \begin{displaymath}c_i(r) = \frac{n_i}{\rho}\left(1-\frac{a}{r}
\end{displaymath} (31)

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 $-4\pi
r^2 \vec{j}^{\rm diff}_i$, according to Eqs. (25) and (31), is

 \begin{displaymath}{\frac{{\rm d}V}{{\rm d}t}}^{\rm sKn} = 4\pi~a \sum\limits_{\...
...}D_{\rm r}
n_{\rm r} \left(1-\frac{1}{S_{\rm r}}\right)\cdot
\end{displaymath} (32)

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).

The general case:

\par\epsfig{file=h3912_3.eps, width=8.8cm} \end{figure} Figure 3: Contour plot of the growth time-scale $\log \tau_{\rm gr}$ [s] as function of the grain radius a and the gas density $\rho $ at constant temperature $T\!=\!1500$ K for quartz (SiO2, $\rho_{\rm d}\!=\!2.65\rm~g~cm^{-3}$). We assume growth by accretion of the key species SiO with solar particle density $n_{\rm SiO} =
10^{(7.55-12)} n_{\rm \langle H\rangle}$ and extreme supersaturation ( $S\!\to\!\infty$). $r_{\rm SiO}=2\times 10^{-8}$ cm is estimated.

For arbitrary Knudsen numbers we apply the same interpolation scheme as outlined in Sect. 2.3. We define a critical Knudsen number $Kn^{\rm cr}$ by equating Eqs. (22) with (32)

\begin{displaymath}a~v^{\rm rel}_{\rm r}~\alpha_{\rm r} = D_{\rm r} .
\end{displaymath} (33)

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

\begin{displaymath}{Kn^{\rm cr}_{\rm r}} = \frac{\bar{\ell}}{2a}
= \frac{3\pi ...
{8~\bar{\sigma}\sqrt{1+\frac{m_{\rm r}}{m_{\rm H_2}}}}\cdot
\end{displaymath} (34)

Assuming perfect sticking ( $\alpha_{\rm r}\!=\!1$) and considering typical molecular radii and masses between $r_{\rm CO}\!=\!1.89$ Å and $r_{\rm H_2O}\!=\!2.33$ Å, and between $m_{\rm CO}\!=\!28$ amu and $m_{\rm H_2O}\!=\!18$ amu, respectively, the resulting critical Knudsen numbers for growth $Kn^{\rm cr}_{\rm r}$ 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

\begin{displaymath}Kn^{\rm cr} ~\approx~ 0.2 .
\end{displaymath} (35)

Our ansatz of the general volume accretion rate for arbitrary Knudsen numbers with $Kn'\!=\!Kn/Kn^{\rm cr}$ is

 \begin{displaymath}\frac{{\rm d}V}{{\rm d}t} ~=~ {\frac{{\rm d}V}{{\rm d}t}}^{\r...
...{\rm d}t}}^{\rm sKn}\!\!
\left(\frac{1}{Kn'+1}\right)^2 \cdot
\end{displaymath} (36)

Figure 3 shows the resulting growth time-scale

\begin{displaymath}\tau_{\rm gr} = \frac{4\pi~a^3}{3}\bigg{/}\frac{{\rm d}V}{{\rm d}t}
\end{displaymath} (37)

as function of particle size a and gas density $\rho $ for the example of quartz grains in an extremely supersaturated gas ( $S\!\to\!\infty$) of solar abundances, where all Si is bound to SiO. We consider the explicit growth reaction

 \begin{displaymath}\rm SiO + H_2O \;\longrightarrow\; \rm SiO_2(s) + H_2 \ ,
\end{displaymath} (38)

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 $\tau_{\rm gr}\propto a/\rho$ whereas for small Knudsen numbers the growth time-scale increases faster for larger grains and becomes density-independent, $\tau_{\rm gr}\!\propto\!a^2$. 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 $\tau_{\rm gr}\!=\!\tau_{\rm sink}$. This line defines a maximum dust grain size $a_{\rm max}$ in a brown dwarf atmosphere. For larger particles ( $a\!>\!a_{\rm max}$ above the dashed line), the growth time-scale exceeds the time-scale for gravitational settling ( $\tau_{\rm gr}\!>\!\tau_{\rm sink}$) 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 $a_{\rm max}$ varies between $\approx$$\mu$m in the thin, outer atmospheric regions ( $\rho\!\approx\!10^{-8}~\rm g/cm^3$) and $\approx$100 $\mu$m in the dense, inner regions ( $\rho\!\ga\!10^{-5}~\rm g/cm^3$). These values depend on the stellar parameters and the considered dust material density.

Note, that an absolute minimum of $\tau_{\rm gr}$ has been considered in Fig. 3 since extreme supersaturation ( $S\!\to\!\infty$) and solar abundance of silicon in the gas phase have been assumed. In the case of an Si-depleted or nearly saturated gas ($S\!\ga\!1$), $\tau_{\rm gr}$ becomes larger and the maximum particle radius $a_{\rm max}$ becomes smaller. Furthermore, the values for $a_{\rm max}$ are relatively independent of temperature, but will shift as $a_{\rm max}\!\propto\!g$ in the free molecular flow case and $a_{\rm max}\!\propto\!\sqrt{g}$ in the laminar viscous case, remembering that $\vec{v}_{\rm dr}^{\hspace{-0.6ex}^{\circ}}\!\propto\!g$ in both cases and that $H_{\rm p}\!\propto\!1/g$ (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 $Re_{\rm d}\!>\!1000$ 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.

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