## Abstract

Atmospheric dust from volcanoes, sand storms and biogenic products provides condensation seeds for water cloud formation on the Earth. Extrasolar planetary objects such as brown dwarfs and extrasolar giant planets have no comparable sources of condensation seeds. Hence, understanding cloud formation and further its implications for the climate requires a modelling effort that includes the treatment of seed formation (nucleation), growth and evaporation, in addition to rain-out, mixing and gas-phase depletion. This paper discusses nucleation in the ultra-cool atmospheres of brown dwarfs and extrasolar giant planets whose chemical gas-phase composition differs largely from the terrestrial atmosphere. A kinetic model for atmospheric dust formation is described, which, in recent work, has become part of a cloud-formation model. For the first time, diffusive replenishment of the upper atmosphere is introduced as a source term into our model equations. This paper further aims to show how experimental and computational chemistry work links into our dust-formation model, which is driven by applications in extraterrestrial environments.

## 1. Introduction

Atmospheric dust provides the condensation seed for water cloud formation on the Earth; hence, it is essential for precipitation and climate evolution. The main sources of such dust on the Earth are volcano ash, sand storms and water sprays. A sand storm from the Sahara dessert can transport 10^{8} t of sand, which translates into 10^{31} silicate particles of size 0.01 μm. Other sources on the Earth include photodissociation and biogenic products, but are of considerably smaller importance. None of these processes, however, reaches above the Earth's mesosphere where noctilucent clouds form at a height of 76–85 km. None of these processes is available in brown dwarfs, giant gas planets or other planetary objects without plate tectonics. Brown dwarfs are very low mass stars with masses below 0.08 M_{⊙}, hence their mass is too low to allow for hydrogen fusion as a long-term internal energy source (for a review, see [1]). After a short period of deuterium burning, brown dwarfs cool and shrink for the rest of their life. This cooling process makes them become more and more like planets with respect to their effective temperature, and hence also with respect to their atmosphere temperatures. It is therefore not surprising that brown dwarfs also form clouds in their atmosphere, as recent transit spectroscopy measurements show [2].

However, puzzles also prevail on crusty objects similar to the Jupiter moon Titan. In October 2004, the Cassini space mission observed a haze layer on Titan that was too high in the atmosphere to be linked to dust from Titan's surface. Liang *et al.* [3] and Lavvas *et al.* [4] demonstrated that photo-chemical reactions led to the formation of macro-molecules, with C_{6}N_{2} being one of the most abundant species. Cassini also revealed a second, detached haze layer in Titan's mesosphere at 520 km where photo-chemical reactions are inefficient; therefore, they cannot serve as an explanation for the occurrence of this haze. Rosinski & Snow [5] and Saunders & Plane [6] followed the idea that condensation seeds for terrestrial noctilucent clouds form as result of meteor evaporation and subsequent condensation of gas-phase metal oxide and silicate species at such height. This idea links to the dust-formation processes from the gas phase in many astronomical environments, such as atmospheres of cool giant stars, brown dwarfs and extrasolar giant planets, as there are no sources of condensation seeds comparable with the Earth or Earth-like objects. The works by Pont *et al.* [2] and Sing *et al.* [7] show that the extrasolar planet HD 189733b also exhibits a haze layer. Using transmission spectroscopy, they demonstrated that its wavelength-dependent size could only be explained by atmospheric dust. Helling *et al.* [8] suggested, using their kinetic dust-cloud-formation model, that such haze layers should exist.

The present paper discusses nucleation (seed formation) in ultra-cool atmospheres and summarizes our model for atmospheric dust formation. The model describes nucleation, growth/evaporation and gravitational settling, and the conservation of each of the involved elements is a set of auxiliary conditions. For the first time, diffusive replenishment is treated in the frame of this model. This paper aims to show how experimental and computational chemistry work would link into our dust model, which is driven by applications in extraterrestrial environments.

## 2. Gas-phase abundances and element conservation

The composition of the gas phase plays an important role for the atmospheric chemistry and, hence, also for the emergent energy spectrum, because the gas provides the molecules, atoms or ions that participate in the condensation process, and because elements will be depleted by dust-formation and will be returned to the gas phase by evaporation. The influence of the dust-formation processes on the gas phase is treated by conservation equations for the elements that participate in the dust-formation processes. The element abundances, *ϵ*_{x}, determine the gas-phase composition *n*_{y} (cm^{−3}). The gas-phase composition is treated in chemical equilibrium by the law of mass action in all that follows in this paper. Figure 1 shows how the element abundances change if dust forms: dust depletes the element abundances below its initial solar values at *T*_{g}<1400 K. An increase in the element abundances occurs if the dust evaporates, see *ϵ*_{Si}, *ϵ*_{Mg}, *ϵ*_{Fe} at *T*_{g}∼1500 K. Hence, the element abundances change with height depending on the dust-formation efficiency. Figure 2 demonstrates how the molecular number densities change with changing element abundances (grey versus blue). The element conservation equation is affected by nucleation (seed formation: first source term), growth and evaporation (mantle growth/evaporation: second source term) as
2.1
with the total hydrogen nuclei density *n*_{〈H〉} (*ρ*=1.424 amu for solar abundances) and the stoichiometric coefficients *ν*_{x,s} of element *x* in solid *s*. Consumption occurs owing to nucleation (solid/liquid species *s*=0) and owing to net growth (reactions *r*=1,…,*R*). See §4*a* for other variable definitions and more details on these processes.

## 3. Dust-formation processes

Dust-formation processes are diverse and not limited to astrophysical environments alone, in particular, if dust–dust processes-like coagulation, sputtering processes and charges are also considered. Therefore, only those processes will be considered in this paper that describe the formation of dust from the gas phase to macroscopic particles. These are the seed formation (nucleation) by gas–gas collisions, the bulk growth by gas–grain collisions and evaporation as the reverse process of surface growth. All these processes act on much shorter time scales than grain–grain collisions and, hence, need to occur first.

Dust formation in atmospheric environments is influenced by gravitational settling (rain), and also by mixing processes comparable to the large-scale Hadley cells on the Earth. Raining dust grains can continue to grow as they drift into regions where condensable material is still present. This process depends strongly on the frictional interaction with the surrounding gas (§5*a*). Convective up-mixing of uncondensed material allows a stationary cloud to develop as it counteracts the depletion of the gas phase by the condensation processes and the subsequent rain-out. Diffusion may pay a similar role (§5*c*).

### (a) Seed formation/nucleation

The formation of a cloud particle (grains) starts with the formation of seed particles. This seed formation proceeds via a net of chemical reactions between gas-phase constituents, and it will eventually lead to an increase in the reaction product with each reaction step. Potentially, a *reaction path* that is the most efficient way through such a *chemical network* can be identified. This path is determined by its slowest reaction (bottle-neck reaction) resulting from the most unstable cluster (critical cluster; [11]). The instability of the critical cluster in comparison with the solid results requires a high supersaturation of the seed-forming gas species (e.g. TiO_{2}) over its solid phase (TiO_{2}[s]). The critical cluster is considerably more unstable than the solid phase, and therefore, a super-cooling below the respective solid's temperature of thermal stability is required to allow the nucleation process to start (figure 3). While thermal stability describes a system's state during which the growth proceeds at the same rate as the evaporation (figure 3), and hence no net effect occurs, the nucleation (and also the mantle growth, see §3*b*) requires the growth rate (*τ*^{−1}_{gr}) to succeed the evaporation rate (*τ*^{−1}_{ev}), and hence leads to a net growth (figure 3*b*(ii)).

Figure 3 further suggests that *the winner takes it all*. Once seeds of one kind start to form, all other thermally stable materials will condense on this surface rather than forming seeds themselves. The question is therefore, if we can identify such a seed-forming species, i.e. such a primary condensate. Figure 4 shows the nucleation rates for four potential seed-forming species (TiO_{2}[s], Al_{2}O_{3}[s], SiO[s], Fe[s]). The element iron is rather highly abundant in an oxygen-rich gas and the atomic iron has a high number density. John & Sedlmayr [14] have shown that the small iron clusters such as Fe_{2} and Fe_{3} have low binding energies and are therefore rather low abundant. Iron is therefore not a good seed-formation candidate. In addition, the element silicon is among the most abundant elements in a solar metallicity gas, and therefore the molecule SiO could be a good candidate to form SiO[s] clusters. Goumans & Bromley [15], however, show that pure SiO nucleation is not feasible. Nuth & Ferguson [16] reach the same conclusion from their updates of vapour pressure measurements. Corundum, Al_{2}O_{3}[s], has often been argued in astrophysics to be a primary condensate, but Patzer *et al.* [17] and Chang *et al.* [18] have shown that it is not the (Al_{y}O_{z})_{N} with *y*=*z* that are abundant, but (Al_{2}O_{2})_{N} clusters are more stable and hence more abundant. This would only allow for a heterogeneous reaction path, not for a homogeneous nucleation process. The element titanium is rather low abundant in a solar metallicity gas compared with Fe and Si, but (TiO_{2})_{N} have high binding energies as demonstrated, for example, by Jeong *et al.* [12]. A comparison with the over-plotted (*T*, *n*_{〈H〉}) profiles for an M-dwarf, brown dwarfs and giant gas planets (Jupiter and Saturn) in figure 4 shows that TiO_{2}[s] nucleates are more efficient than SiO[s]. This suggests that TiO_{2}[s] can be identified as the primary condensate also for brown dwarfs and giant gas planet atmospheres in addition to asymptotic giant branch (AGB), as suggested in Jeong *et al.* [13]. The work by Jeong *et al.* enable us to treat the TiO_{2} nucleation as a homogeneous seed formation that allows an efficient treatment in later atmosphere or hydrodynamic simulations. We note, however, that the TiO_{2}-cluster data have been updated in the literature [19] and that heterogeneous seed formation of silicates is suggested in Saunders & Plane [6].

### (b) Growth of grain bulk

The growth of the bulk dust volume and dust mass only occurs after the seed particles are available for chemical surface processes. As seed formation implies a strong cooling below the thermal equilibrium temperature (*T*<*T*(*S*=1)), many materials are thermally stable when the first seed surface is formed (figure 4). Hence, these particles grow almost simultaneously on top of the seeds and form the so-called core–mantle (or dirty) grains.

The bulk growth/evaporation is influenced by hydrodynamical motion in the dust-forming environment if the growth/evaporation time scales are of the order of hydrodynamic time scales. This can easily be the case as the examples of AGB-star winds and brown dwarf cloud-formation show. The material-dependent radiation pressure on dust grains causes a drift motion of the dust compared with the gas in AGB stars. Strong gravity causes a drift of the falling cloud particles in brown dwarf and planetary atmospheres. Processes counteracting such a rain-out are convection and turbulence, which provide a large-scale, Hadley-cell-like velocity field in the case of convection and a small-scale diffusion process in the case of turbulence. Both processes, drift and diffusion, would ideally be incorporated in a dust-formation formalism as source or sink terms for the respective dust-formation processes.

## 4. A kinetic model for dust nucleation

### (a) Basic description

The most general description of the formation of solid particles or liquid droplets from a gas phase is given by the time evolution of the local size distribution of the particles/droplets. Balancing the source and sink processes allows formulation of a conservation equation [20]. Ideally, one would account for arbitrary chemical reactions that lead to the growth of the clusters, but chemical data would be required for each possible chemical reaction. Therefore, this method is limited to available data on cluster growth and evaporation, and therefore a homogeneous reaction path will be considered.

Classical nucleation theory is widely used in astronomical environments, but it can only be applied if [21]

— the free energy of formation of a cluster of size

*N*,*ΔG*(*N*), can be approximated by macroscopic parameters such as the surface tension of the bulk solid;— the clusters are in thermal equilibrium, hence, the cluster's vibrational temperature and the gas temperature are the same; and

— chemical equilibrium holds for small clusters and in the gas phase.

At low supersaturation, the results of the classical nucleation theory agree well with similar experiments [22]. The formation of raindrops in air is such a process.

Let us consider the formation of solid particles that are made of one kind of monomer (the smallest unit the particles are made of). Each of the particles of size *N* has a number density *f*(*N*,*t*) at each time *t*.

#### (i) Master equation

The conservation equation for the particle density *f*(*N*,*t*) of a grain of size *N* is
4.1
Here, *I* denotes the maximum molecular *i*-mer contributing, the *i*th geometrical configuration of a molecule, to the chemical growth of the particle. *J*^{c}_{i}(*N*,*t*) is the effective flux (or transition rate) for the growth of the particle of size *N*−*i* to size *N* due to all chemical reactions associating or dissociating an *i*-mer of the condensing species. This flux through cluster space is given by
4.2
Equation (4.2) sums over all chemical reactions *r*_{i} in which an *i*-mer is involved. The homogeneous nucleation process is then described by one of these *R*_{i} reactions. *τ*_{gr}(*r*_{i},*N*−*i*,*t*) is the growth time of the growth reaction *r*_{i} leading from the cluster size *N*−*i* to cluster size *N*. *τ*_{ev}(*r*_{i},*N*,*t*) is the evaporation time leading from size *N* to size *N*−*i*.

#### (ii) Growth and evaporation rates and reference equilibrium state

The growth and evaporation rates are expressed by
4.3
and
4.4
with *n*_{f}(*r*_{i}) and *n*_{r}(*r*_{i}) the number density of the molecule of the growth (forward) process and of the evaporation (reverse) process for reaction *r*_{i}, respectively. Equation (4.4) applies to reaction *A*+*B*_{N}→*AB*+*B*_{N−1}. The normal evaporation process would be spontaneous such that *B*_{N}→*B*+*B*_{N−1} and, hence *v*_{rel} does not enter equation (4.4). *α*(*r*_{i},*N*−*i*) and *β*(*r*_{i},*N*) are the reaction efficiency for growth and evaporation via reaction *r*_{i}, *v*_{rel} is the average relative velocity between the growing/evaporating molecule and the cluster. *A*(*N*−*i*) is the surface of the cluster of size *N*−*i*.

The challenge is that the growth and evaporation efficiency coefficients, *α*(*r*_{i},*N*) and *β*(*r*_{i},*N*), are often unknown for the different cluster sizes *N*. The recent work by, for example, Saunders & Plane [6] may provide these data for silicate nucleation.

For further considerations, we need to introduce a *reference equilibrium state* in which the clusters are considered to be in equilibrium with the saturated vapour in which the phase equilibrium between the monomers and the bulk solid/liquid phase holds, hence *S*_{1}=1 (*S*_{1}=*p*_{1}/*p*_{vap,s}—supersaturation ratio of the monomer). Patzer *et al.* [21] show in their appendix A that if the temperatures of all components are equal, the supersaturation ratio of a cluster of size *N* with respect to the bulk is *S*_{N}=(*S*_{1})^{N}. Hence, phase equilibrium between monomers and between the clusters and the bulk solid, plus simultaneous chemical equilibrium in the gas phase, plus thermal equilibrium (i.e. all components have the same temperature) are characterizing this equilibrium state.

In such local thermodynamic equilibrium (LTE) between the gas phase and the clusters, the principle of detailed balance holds for a single microscopic growth process and its respective reverse, i.e. evaporation, process. This implies that under the condition of detailed balance, *f* ° (*N*−1)/*τ*_{gr}(*N*−1)=*f* ° (*N*)/*τ*_{ev}(*N*), which allows expression of the evaporation rate by the growth rate,
4.5
where *f* ° (*N*−*i*), *f* ° (*N*), *n*^{°}_{f}(*r*_{i}) and are the equilibrium particle densities for the clusters of size *N*−*i* and *N*, and of the monomers involved in the growth (*f*) and in the evaporation (*r*) reaction. The law of mass action links these equilibrium particle densities to Gibbs energies, which can be determined from chemical modelling,
4.6
where (kJ mol^{−1}) is the Gibbs free energy of formation and can be calculated from the standard molar Gibbs free energy of formation of all reaction participants at the temperate *T*_{d}(*N*), which in LTE is *T*_{d}(*N*)=*T*_{g},
4.7

Therefore, to determine the rate of nucleation, we need

— the growth time scale

*τ*_{gr}from chemical kinetic theory or rate measurements and— the thermodynamic properties of the

*N*clusters and their reaction partners contained in from equilibrium thermodynamics.

#### (iii) Stationary nucleation rate

If the growth time to macroscopic particles of sizes *N*≥*N*_{l}, which are big enough to show bulk solid properties already, is small compared with hydrodynamic changes in the environment, the time to establish a stationary flow through the cluster space up to size *N*_{l} is negligible. A stationary flow through the cluster space implies a constant (cluster) particle density in time, hence d*f*(*N*,*t*)/d*t*=0 in equation (4.1) and
4.8
Here, *J*_{*}(*t*) is the size-independent rate of cluster formation, or the stationary nucleation rate. Gail & Sedlmayr [20] have shown that the system of principle infinite number of equations in equation (4.8) can be truncated at a suitable cluster size *N*<*N*_{l}. If now the nucleation process is dominated by one or only a few molecules (e.g. C_{2}H_{2} during soot formation), an analytic expression for *J*_{*} can be obtained. For further details, including gas-phase non-equilibrium processes, we refer to Patzer *et al.* [21].

### (b) Nucleation model

We are seeking an analytic expression for the stationary nucleation rate *J*_{*}(*t*). For that, we summarize the complex work presented in Patzer *et al.* [21] for those expressions that we need to model dust nucleation in brown dwarfs and planetary atmospheres. A similar ansatz is used for modelling the formation and evolution of dust-driven winds of AGB stars [23,24,25].

We describe the seed-formation process as a homogeneous, homomolecular process during which the stepwise addition of (only) one molecule in each reaction step leads to the formation of larger and larger clusters. We assume that such a homogeneous reaction chain can be identified. This assumption is made for the sake of numerical efficiency [21].

The slowest reaction will determine the flux through the cluster space during such a chain of reaction. This bottle-neck reaction will lead to the formation of the critical cluster, *N*_{*}, which is the least stable cluster and once it has formed, constructive cluster growth processes will dominate. This means that *J*_{*}(*t*) is determined by the quantities of the critical cluster. The stationary nucleation rate for a homogeneous, homomolecular process in thermal equilibrium with the gas phase (*T*_{d}(*N*)=*T*_{g}=*T*) is therefore,
4.9
The equilibrium cluster size distribution is expressed by a Boltzmann-like distribution,
4.10
with *f* ° (1) the equilibrium density of the monomer. If the monomers are in phase equilibrium with the solid phase (*S*=1), which is also in thermal equilibrium with the gas (*T*_{d}(*N*)=*T*_{g}), and if the gas phase is in chemical equilibrium including the *N*-mers and monomers, than the *N*-mers are in phase equilibrium with the solid, too. Therefore,
4.11
with *S*_{N} the supersaturation ratio for the *N*-mer (cluster of size *N*).

*ΔG*(*N*) is the free-energy change owing to the formation of a cluster of size *N* from the saturated vapour. It is related to the standard molar Gibbs free energy of formation of the *N* cluster by
4.12
where is the standard molar Gibbs free energy of formation of the solid phase, and *p*^{°−} is the pressure of the standard state. Most often, *p*^{°−} is the atmospheric pressure on the Earth at which *p*_{sat} and *Δ*^{°−}_{f} *G*(*N*,*T*) were measured. The right-hand side of equation (4.12) contains new quantities, which can be determined from laboratory experiments or quantum- chemical calculations.

Applying equations (4.10) and (4.11) to equation (4.9) results in an analytic expression for the stationary nucleation rate,
4.13
Borrowing from the classical nucleation theory (see §4*a*), we consider the clusters to be spherical droplets with a hypothetical monomer radius *a*_{0}, and with *σ* being the surface energy of the cluster of size *N*,
4.14
Here, *N*_{f} is a fitting factor representing the particle size at which the surface energy is reduced to half of that of the bulk value. The representation of *ΔG*(*N*) in equation (4.14) yields the correct limit and it accounts for the effect of the curvature on the surface energy for small clusters [26].

The critical cluster size *N*_{*} is given by
4.15
and the corresponding Zeldovich factor is
4.16

In the frame of the *classical nucleation theory*, equation (4.14) simplifies for *N*≫1 to
4.17
Hence, the term *ΔG*(*N*)/(RT) reduces to a dependence on the supersaturation ratio, *S*, on the surface tension, *σ* (contained in ), only. Inserting this into equation (4.10) and both into *J*_{*}(*t*)=(*f* ° (*N*_{*})/*τ*_{gr}(1,*N*_{*},*t*))⋅*Z*(*N*_{*}) leads to the result of the classical nucleation theory.

Jeong *et al.* [12] have now used computational data for (TiO_{2})_{N} clusters to provide a better representation for the surface energy *σ* (erg cm^{−2}). With that, they found for the classical expression for *ΔG*(*N*) in equation (4.17) that *σ*=521 erg cm^{−2}, and for the modified expression for *ΔG*(*N*) in equation (4.14) that *σ*=618 erg cm^{−2} with *N*_{f}=0. Updated cluster data suggest an update of these values.

## 5. Kinetic model for bulk growth/evaporation

Once the seed formation has set in, surface growth will follow and produce the bulk volume and the bulk mass of the atmospheric dust forming from the gas phase. This particle growth (and evaporation) is by chemical surface reactions.

Re-writing the master equation (equation (4.1)) in the Eulerian frame for dust particles ∈ [*V* , *V* +*dV* ] in the following way will allow us to consider particle growth beyond the nucleation regime:
5.1
*f*(*V*) (cm^{−6}) is the distribution function of grains in volume space, which accounts for different monomer volumes for different chemical reactions. The right-hand side of equation (5.1) expresses the population and depopulation of the considered volume interval [*V* , *V* +d*V* ] with dust particles that are changing their size due to accretion or evaporation of molecules or atoms (figure 5)
5.2
Multiplication of equation (5.1) by *V* ^{j/3} (*j*=0,1,2,…) and integration over *V* from a lower integration boundary *V* =*V* _{ℓ} to results in
5.3
where the *j*th moment of the dust size distribution function *L*_{j} (cm^{j}/g) is defined by
5.4
The source term on the left-hand side of equation (5.3) expresses the effects of seed-formation and surface chemical reactions on the dust moments. Other additional source terms could comprise the effects caused by a size-dependent drift motion of the grains, e.g. due to gravity, or by diffusion.

Consideration of the atmosphere of a brown dwarf or giant gas planet with a gas-phase chemistry that contains more oxygen than carbon (figure 2) poses the following challenges.

— The growing/evaporating grain moves in an environment of changing density and gas–grain interaction and therefore changes between a free molecular flow to a diffusive process. This influences the drift velocity (gravitational settling) and the growth speed.

— The oxygen-rich gas is chemically very diverse, leading to many different surface reactions that can contribute to the formation of various condensates.

— Large-scale convection and/or small-scale diffusion due to turbulence replenish the dust-forming atmospheric regions with uncondensed material and possibly condensation seeds.

### (a) Growth, evaporation and drift

Gravity causes cloud particles to fall down. This gravitational settling determines the vertical extension of the cloud. The dust and the gas move with different velocities such that . The master equation for dust particles ∈[*V*,*V* +d*V* ] as introduced in equation (5.1) is then expressed in equation (5.5) from which the modified dust moment equation (5.6) follows:
5.5
and
5.6
The source term expresses the effects of seed formation and surface chemical reactions on the dust moments and is an additional, advective term that comprises the effects caused by a size-dependent drift motion of the grains, e.g. due to gravity.

The relative velocity between dust and gas, , depends on the grain size and the local gas density. If the mean free path of the gas is larger than the grain size, the Knudsen number *Kn*=*l*/1*a*≫1, the frictional force for rarefied gases is applied. If *Kn*≪1, the viscous case needs to be considered (for details, see [24]). These changing gas-flow regimes also influence the growth of the grains by surface reactions.

#### (i) Free molecular flow (*Kn*≫1)

For large Knudsen numbers, 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 that causes a growth step (or an evaporation step) of the dust particle.

The accretion rate, expressed in terms of the increase of the particle's volume *V* =4*πa*^{3}/3 due to chemical surface reactions for large Knudsen numbers, is given by
5.7
where *r* is a general surface reaction index, *V* _{r} is the increase of the dust particle's volume *V* caused by one reaction *r*. Here, *n*_{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 determined from the particle densities present in the ambient gas and stoichiometric considerations. Strictly speaking, the key educt is identified by the minimum among all educts of reaction *r*, where is the stoichiometric factor of educt *k* in reaction *r*. The relative velocity is here defined as , where *m*_{r} is the mass of the key species. *α*_{r} is a sticking coefficient that can contain more detailed knowledge about the surface chemical process, if available [27]. The sticking coefficient is the ratio between physisorption rate and thermal collision rate.

The last term on the right-hand side of equation (5.7) takes into account the reverse chemical whether the dust particle grows or shrinks. is a generalized supersaturation ratio of the surface reaction *r* [28], where is the particle density of the key species in phase equilibrium over the condensed dust material. In the case of simple surface reactions, which transform *m*_{r} units of the solid material from the gaseous into the condensed phase (and vice versa), e.g. (, *m*_{r}=1), the generalized supersaturation ratio is
5.8

In the subsonic case (), the growth (↑) and evaporation (↓) rates are
5.9
5.10
5.11
and
5.12
Here, detailed balance considerations (Milne relation), thermal equilibrium (*T*_{d}=*T*_{g}=*T*) and chemical equilibrium are applied (see §5*b*).

The frictional force in the subsonic case results in an equilibrium drift velocity of
5.13
where *a* is the particle radius, *ρ*_{d} the dust material density, *e*_{z} the vertical unit vector (pointing upwards) and *g* the gravitational acceleration (downwards). is the mean thermal velocity with *T* being the temperature, *k* the Boltzmann constant and the mean molecular weight of the gas particles. By means of equations (5.9)–(5.13), the integrals on the right-hand side of equation (5.6) can be evaluated as shown in more detail concerning the term in [20]. After some algebraic manipulations, including *a*=(3*V*/4*π*)^{1/3}, the approximation *ΔV* _{r}≪*V* and partial integration, the results are
5.14
and
5.15
where *J*(*V* _{ℓ})=*f*(*V* _{ℓ})(d*V* /d*t*)|_{V =Vℓ} is the current of dust particles in volume space at the lower integration boundary. In case of net growth, can be identified with the stationary nucleation rate *J*_{*} [20]. The characteristic growth speed (cm s^{−1}) (an increase in radius per time) and the characteristic gravitational force density *ξ*_{lKn}(dyn cm^{−3}) are given by
5.16
and
5.17

#### (ii) Viscous case (*Kn*≪1)

For small Knudsen numbers, 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 equation (5.7), but is hindered by inter-molecular collisions. Consequently, the gain growth and evaporation is limited by the diffusion of the molecules, towards or away from the grain's surface considering growth or evaporation, respectively.

The total volume accretion rate of a grain for small Knudsen numbers, summing up the contributions of several surface reactions of index *r* (as in equation (5.7)) with the net rate −4*πr*^{2}*j*^{diff}_{i} according to Woitke & Helling [27], is
5.18
*D*_{r} is the diffusion constant of the key reaction species related to reaction *r* (see table 1 in [29]) in a gas mainly composed of H_{2} molecules,
5.19
where is the thermal velocity of a gas particle *i* with reduced mass 1/*m*_{red}=1/*m*_{H2}+1/*m*_{i}, where *m*_{H2} and *m*_{i} are the masses of H_{2} and *i*, respectively. is the total gas particle density. The description of the volume accretion rate according to equation (5.18) allows for the simultaneous growth of different solid materials on the same surface (heterogeneous growth). Note that *r*_{H2} and *r*_{i} are the radii of the respective molecules in equation (5.19) (not to be confused with the reaction number *r* as in equations (4.2)–(4.7)).

For a laminar flow (Reynolds number *Re*_{d}<1000), the derivation of the moment equations for the case of small Knudsen numbers is analogous to the previous subsection. We express the surface chemical rates according to equation (5.18) by
5.20
5.21
5.22
and
5.23
assuming thermal and chemical equilibrium. The equilibrium drift velocity for a laminar flow and *Kn*≪1 is
5.24
By repeating the procedure of the last subsection, the right-hand side terms of the dust moment equations are
5.25
and
5.26
The characteristic growth speed now has the units (cm^{2} s^{−1}) (an increase in surface per time), whereas the characteristic gravitational force density *ξ*_{sKn} remains the same, except for a different geometry factor,
5.27
and
5.28

### (b) Formation of dirty dust grains

So far, the surface growth/evaporation was treated for homogeneous growth of one material at the time only. The thermal stability of oxygen-rich materials suggest that metal oxides such as SiO[s], FeO[s], MgO[s] and silicates could grow simultaneously onto a seed particle. For quantitative modelling of the surface growth and evaporation, we assume that the total grain serves as a funnel to accrete the gaseous species, and that the solid's evaporation process proceeds only from the surface of the islands of its own kind. These islands are assumed to be equally distributed on the surface as well as in the total volume of the grain, which will be denoted by ‘well-mixed’ dust grains.

For a subsonic flow with a larger Knudsen number, the growth and evaporation rates, which cause a (de-)population of the considered volume interval [*V*,*V* +*ΔV* ], can be expressed by
5.29
5.30
5.31
and
5.32
*A*_{tot}(*V*) denotes the total surface area of the grain and is the integrated surface area of all islands of solid *s* (*i* enumerates the islands of kind *s*). *α*_{r} is the sticking coefficient (ratio between physisorption rate and thermal collision rate) and *β*_{r} the evaporation rate coefficient of reaction *r*. *n*_{r} is the particle density of the key educt, which is the least abundant among the reactant molecules *k*=1…*K*, and *v*^{rel}_{r} its thermal relative velocity. Each reaction *r* leads to an increase (or decrease) in the dust grain volume by *ΔV* _{r}.

Applying the concept of detailed balance (Milne relation), which is valid for each individual reaction *r*, we eliminate the evaporation rate coefficient *β*_{r}. This reference state, which will be denoted by ^{°}, is characterized by phase equilibrium with the considered solid (*S*=1), which is in a pure state (*A*_{s}=*A*_{tot}) and has an infinite flat surface. Furthermore, thermal equilibrium between gas and solid (*T*_{g}=*T*_{d}), and chemical equilibrium in the gas phase () are valid.

Using these assumptions, the evaluation of the right-hand side of equation (5.3) for sufficiently large (*V* ≥*V* _{ℓ}≫*ΔV* _{r}) and spherical particles () yields, with partial integration
5.33
The last factor in equation (5.33) can be expressed by means of the following identity:
5.34
which is the ratio between the growth and the evaporation reaction rates of a pure solid. The supersaturation ratio *S* and the non-equilibrium *b* factors are defined as
5.35
5.36
5.37
and
5.38
Equations (5.35)–(5.38) express the effects of the different types of non-equilibria on the ratio between growth and evaporation rates. The explicit formulae for the supersaturation ratio *S* and the *b* factors *b*_{therm} and *b*_{chem} are only valid for type I surface reactions.

Different materials *s*=0…*S* have different monomer volumes *ΔV* _{s}, and the number of monomers of the solid *s* is *N*_{s}. The total dust volume, *V* , and the volume fraction, *b*^{s} of the material *s* are then given by
5.39

The total dust volume per cubic centimetre stellar matter, *V* _{tot}, is given by the third dust moment
5.40
where *ρ* (g cm^{−3}) is the mass density and *V* _{ℓ} is the lower integration boundary. Similarly, we define the volume *V* _{s} of a certain solid species *s* by
5.41
where *V* ^{s} (cm^{3}) is the sum of island volumes of material *s* in *one* individual dust particle. For simplicity, we assume that *V* ^{s}/*V* =*V* _{s}/*V* _{tot} is constant for all dust particles at a certain position in the atmosphere, i.e. we assume a unique volume composition of all grains at (** x**,

*t*), such that and .

By means of this assumption, it is possible to express the integrals that occur after integrating the master equation (5.5) over size in terms of other moments. The results are the dust moment conservation equations. The change in the partial volume of the solid *s* can then be expressed analogously to eqn (23) in Helling & Wotke [30],
5.42
where we have assumed that the Knudsen numbers are large (*Kn*≫1) and that the drift velocities **v**_{dr}(*V*) of the dust particles can be approximated by the equilibrium drift velocities (final fall speed).

The source terms on the right-hand side of equation (5.42) describe the effects of nucleation, growth and evaporation of condensate *s*. (cm^{3}) is the volume occupied by condensate *s* in the seed particles when they enter the integration domain in size space. The net growth velocity of condensate *s*, *χ*^{s}_{net} (cm s^{−1}) (negative for evaporation), is (for more details, see [30])
5.43
Here, *r* is again the index for the chemical surface reactions, is the volume increment of solid *s* by reaction *r* (), is the particle density of the key reactant, is its thermal relative velocity and *α*_{r} is the sticking coefficient of reaction *r*. *S*_{r} is the reaction supersaturation ratio and is a *b* factor (equation (5.39)) that describes the probability of finding a surface of kind *s* on the total surface. Putting independent of *V* , we assume that all grains at a certain point in the atmosphere have the same surface and volume composition, i.e. the grain material is a homogeneous mix of islands of different kinds.

The divergence of the drift term in equation (5.42) is treated in the following way. Assuming large Knudsen numbers and subsonic drift velocities, the equilibrium drift velocity is given by equation (5.13). The dust material density is now calculated from , which is the dirty dust material density, and *ρ*_{s} is the material density of a pure condensate *s*. Inserting this formula into the drift term in equation (5.42) yields
5.44
with the abbreviation . Defining , equation (5.42) in this paper can be rewritten as
5.45
Equation (5.45) describes the evolution of the partial dust volume of solid *s* in space and time due to advection, nucleation, growth, evaporation and drift.

The third moment equation for the total dust volume *ρL*_{3} (eqn (1) in [31]) can be retrieved by summing the contributions from all condensates *s* as given by equation (5.45), because
5.46

### (c) Future developments: diffusive source terms

The formation of a cloud is described by the nucleation process that determines the number of cloud particles formed, the growth/evaporation processes that determine the total dust mass and volume and the gravitational settling that supports further grow and determines the geometrical extension of the cloud. To allow a cloud to survive longer than the fall time of its cloud particle, material must be brought up into the dust-forming regions. Large-scale convection in the form of strong winds or diffusion processes that are much slower can provide replenishment. Convective up-mixing is considered in Woitke & Helling [31], and an overview of the state-of-the-art in the present model atmospheres is given in Helling *et al.* [32]. In this paper, we introduce diffusive mixing into our dust-formation model.

Diffusive mixing leads to an additional source term, ∇⋅*j*_{dust}d*V* , in the master equation (4.1) that contains the diffusive flux *j*_{dust},
5.47
The resulting dust moment equation is then
5.48
The terms and are given by equations (5.14), (5.15), (5.25) and (5.26), and the diffusive flux is defined as
5.49
where *D*_{d} is the diffusion constant for the dust particles and is expressed according to Schräpler & Henning [33] as a function of the gas diffusion constant, *D*_{g}, and the dimensionless Stokes number, *St*, as
5.50
The stopping time, *τ*_{stop}, depends on the drift velocity, and hence, changes depending on the Knudsen number regime from a free molecular flow to a viscous case (equations (5.13) and (5.24)). The turbulent time scale, *τ*_{eddy}, can represent a slow, nearly laminar flow or a vivid, turbulent flow. An analysis of the Stokes number for different atmospheric profiles with different local temperatures, gas densities and grain properties reveals that different regimes occur for high and low atmospheric density regions (figure 6). For a giant gas planet's atmosphere with, for example, an effective temperature of *T*_{eff}=1600 K and a surface gravity of in figure 6, the consideration of *St*≪1 might be sufficient.

A formal separation of variables in the Stokes number, 5.51 was applied to make use of the volume integral in the moment equations. Applying the dust diffusion coefficient as defined in equation (5.50), the diffusive source therm of the dust moment equations (5.48) can be written as 5.52 5.53 5.54 and 5.55 with the integrals 5.56 and 5.57 Since this form does not allow for direct moment expression, we will analyse different Stokes number regimes.

#### (i) *St*≫1 regime

We re-write the integrals in equations (5.56) and (5.57) by applying the first-order approximation 1/(1+*St*)≈1/*St*,
5.58
5.59
and
5.60
5.61
5.62

Substituting the integrals *I*_{1} and *I*_{2} into equation (5.55), we obtain
5.63

#### (ii) *St*≪1 regime

Now we re-write the integrals in equations (5.56) and (5.57) by applying the second-order Taylor expansion 1/(1+*St*)≈(1−*St*+*St*^{2}) and 1/(1+*St*)^{2}≈(1−2*St*+3*St*^{2}), which results in
5.64
5.65
and
5.66
5.67

Substituting the integrals *I*_{1} and *I*_{2} for *St*≪1 into equation (5.55), we obtain
5.68
5.69
Equation (5.69) is now the first-order approximation of in the small Stokes number regime. The diffusive source term for small Stokes numbers requires a closure condition as it depends on *L*_{j+α}, in contrast to the large Stokes number case. It remains to be seen how efficient the moment equations can be solved numerically if diffusion is taken into account.

Figure 6 shows the Stokes numbers (black lines) for results from Drift-Phoenix model atmosphere simulations [9], which provide gas density and mean grain sizes according to the above dust model. This figure demonstrates that a simple one-parameter diffusion mixing ansatz is not sufficient to describe mixing in a cloud-forming atmosphere. Five atmospheric Stokes regimes can be distinguished in the density grain size space. However, the density values given below should only be used as a guidance, and the exact values will vary for different model atmospheres.

— :

*no dust diffusion*because*St*≫1 in the uppermost atmosphere. The dust is much larger than the gas and the density of the atmosphere is very small.— :

*upper asymptotic part of transition regime*. First-order Taylor expansion of dust diffusion for*St*≫1 is applicable, resulting in equation (5.63).— :

*transition regime*. In this region, diffusion of dust of different sizes has different behaviours, and therefore cannot be described by one set of momentum equations. Interpolation of both asymptotic regimes,*St*≪ 1 and*St*≫ 1, may be needed.— :

*lower asymptotic part of transition regime.*First-order Taylor expansion of dust diffusion for*St*≪1 is applicable, resulting in equation (5.69). It is important to note that this equation has two terms, where the first term is due to dust diffusing like gas, and second is the first-order correction to it.— :

*in the inner atmosphere,*the Stokes number is so small that*D*_{d}=*D*_{g}. Hence, diffusion is described only by the first term of equation (5.69).

## 6. Conclusion

This paper summarizes our model of dust formation, which includes homogeneous nucleation, growth and evaporation of heterogeneous dust grains, and element conservation. It contains recent developments with respect to the gravitational settling, which is essential for describing the formation of a large-scale cloud structure. An outlook is given regarding future developments and a first suggestion is made of how to include diffusive transport terms into dust moment equations.

The applicability of our model equations to astrophysical environments like brown dwarfs or planetary atmospheres requires a certain amount of input data including chemical reaction paths for nucleation, the Gibbs free energy of formation for clusters (equation (4.12)), surface reactions and their efficiencies (e.g. equations (5.9)–(5.12)) for similar and different material growth/evaporation, and diffusion coefficients (equations (5.19) and (5.50)). This need for input data links astrophysical dust formation with laboratory experiments and with efforts in computational chemistry to understand the formation and structures of large clusters as part of cloud formation.

## Acknowledgements

C.H. thanks C. Bilger for providing figures 1 and 2. C.H. highlights financial support of the European Community under the FP7 by an ERC starting grant. Most of the literature search has been performed using ADS.

## Footnotes

One contribution of 11 to a Theme Issue ‘Surface science in the interstellar medium’.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.