Fully time-dependent cloud formation from a non-equilibrium gas-phase in exoplanetary atmospheres

Recent observations suggest the presence of clouds in exoplanet atmospheres but have also shown that certain chemical species in the upper atmosphere might not be in chemical equilibrium. The goal of this work is to calculate the two main cloud formation processes, nucleation and bulk growth, consistently from a non-equilibrium gas-phase. The aim is further to explore the interaction between a kinetic gas-phase and cloud micro-physics. The cloud formation is modeled using the moment method and kinetic nucleation which are coupled to a gas-phase kinetic rate network. Specifically, the formation of cloud condensation nuclei is derived from cluster rates that include the thermochemical data of (TiO$_2$)$_N$ from N = 1 to 15. The surface growth of 9 bulk Al/Fe/Mg/O/Si/S/Ti binding materials considers the respective gas-phase species through condensation and surface reactions as derived from kinetic disequilibrium. The effect of completeness of rate networks and the time evolution of the cloud particle formation is studied for an example exoplanet HD 209458 b. A consistent, fully time-dependent cloud formation model in chemical disequilibrium with respect to nucleation, bulk growth and the gas-phase is presented and first test cases are studied. This model shows that cloud formation in exoplanet atmospheres is a fast process. This confirms previous findings that the formation of cloud particles is a local process. Tests on selected locations within the atmosphere of the gas-giant HD 209458 b show that the cloud particle number density and volume reach constant values within 1s. The complex kinetic polymer nucleation of TiO$_2$ confirms results from classical nucleation models. The surface reactions of SiO[s] and SiO$_2$[s] can create a catalytic cycle that dissociates H$_2$ to 2 H, resulting in a reduction of the CH$_4$ number densities.

In collisionally dominated environments, the gas-phase can be modeled using chemical equilibrium models (Stock et al. 2018;Woitke & Helling 2021).This is a good assumption for the deep atmosphere of exoplanets and brown dwarfs (Venot et al. 2018) where the collisional timescales become small.In low density environments, like the outflow of AGB stars (Plane & Robertson 2022;Sande & Millar 2019) and the upper atmospheres of exoplanets (Rimmer & Helling 2013;Baxter et al. 2021;Tsai et al. 2023;Mendonça et al. 2018) and brown dwarfs (Helling & Rimmer 2019;Lee et al. 2020), collisional timescales become large.In these environments, chemical disequilibrium processes, like radiation or quenching, can drive the gas-phase abundances out of equilibrium.
The formation of clouds and dust starts with the formation of cloud condensation nuclei (CCNs).In gaseous exoplanets, CCNs cannot originate from the planet's surface, like in terrestrial planets, but have to be formed directly from the gas-phase through nucleation, which marks the transition from gas-phase chemistry to solid-phase chemistry.Classical nucleation theory (CNT) or modified classical nucleation theory (MCNT) is an often used approach to describe the rate at which CCNs are formed (nucleation rate).In order to describe the actual formation of clusters leading up to CCNs, a kinetic description can be used.(Patzer et al. 1998;Lee et al. 2015;Bromley et al. 2016;Boulangier et al. 2019;Köhn et al. 2021;Gobrecht et al. 2022).To calculate kinetic nucleation, the thermodynamic properties of clusters of the nucleating species have to be known.There are active efforts to derive the structures and properties of species that are associated with nucleation processes (Chang et al. 2005(Chang et al. , 2013;;Patzer et al. 2014;Lee et al. 2015;Gobrecht et al. 2022;Sindel et al. 2022;Andersson et al. 2023).Even in the cases where data is available, it is often limited to the smallest cluster sizes because calculating the required properties becomes more computationally intensive for larger clusters (Sindel et al. 2022).
Supersaturated species can grow onto CCNs once they are present.There are two main ways these bulk growth processes can occur.Firstly there is condensation, which describes the deposition of gas-phase species onto CCNs (e.g.SiO → SiO[s]).Many models use condensation curves to determine where clouds can form (e.g.Demory et al. 2013;Webber et al. 2015;Crossfield 2015;Kempton et al. 2017;Roman & Rauscher 2017;Roman et al. 2021).Secondly, bulk growth can occur through kinetic surface reactions (Patzer et al. 1998;Helling & Woitke 2006;Helling & Fomins 2013).In contrast to condensation, surface reactions include multiple chemical species to form the bulk material (e.g.SiS + H 2 O → SiO[s] + H 2 S ).In addition to providing additional bulk growth paths for condensing species, surface reactions also allow the bulk growth of materials which may not be stable in the gas-phase themselves.Both processes can be described kinetically (Patzer et al. 1998).
In this paper, we present a fully time-dependent description of the processes that lead to the formation of clouds in exoplanets, or dust in brown dwarfs, stars, and supernovae.We advance the nucleation description by modeling potential chemical pathways based on thermodynamic cluster properties and expand the kinetic description of surface reactions to chemical disequilibrium.With this model, we study the timescales of cloud formation in exoplanet atmospheres.The description of the chemical network, kinetic nucleation and bulk growth is given in Sect. 2. The kinetic chemistry and kinetic nucleation are investigated in Sect.3. The fully kinetic cloud formation model is then applied to temperature-pressure (T gas -p gas ) points within the atmosphere of HD 209458 b (Sect.4).Lastly, the summary is given in Sect. 5.

Model
We present the models for the gas-phase, kinetic nucleation and bulk growth through condensation and surface reactions.In Sect.2.1, we describe our chemical kinetics network of the gas-phase.The kinetic nucleation, which describes the timedependent nucleation within disequilibrium environments, is described in Sect.2.2 (two-body reactions) and in Sect.2.3 (three-body reactions).In Sect.2.4, we describe the bulk growth through condensation and surface reactions.The derivation of the reaction supersaturation is given in Sect.2.5.Finally, the connection between nucleation and bulk growth is described in Sect.2.6.

Gas-phase chemistry
The evolution of the number density n i [cm −3 ] of a given species i is determined by the following equation: where F i the set of reactions where the i-th species is a product, D i the set of reactions where the i-th species is a reactant, E r the set of reactants of reaction r, n j [cm −3 ] the number densities of the reactants, ν i,r the stoichiometric coefficient of the i-th species within reaction r, k r [cm 3(J r −1) s −1 ] the reaction rate for the reaction r and J r the number of reactants in E r .The sum over all n j is the total number density.The numerical solver is described in Appendix A. Chemical kinetic networks for the atmospheres of exoplanets include several hundred species and several thousand reactions (e.g.Rimmer & Helling 2016;Tsai et al. 2017Tsai et al. , 2021;;Venot et al. 2012Venot et al. , 2020)).For this paper, we chose the "NCHO thermo network" of VULCAN 1,2 (Tsai et al. 2017(Tsai et al. , 2021)).This network includes 69 species and 780 reactions.In this work, TiO 2 is considered as the nucleation species.Because "NCHO thermo network" of VULCAN does not include reactions for the formation of TiO 2 , we add several reactions from Boulangier et al. (2019) leading to the formation of TiO 2 .The selected gas-phase reactions can be found in Table E.1.Furthermore, we add reactions including Si species for the bulk growth species SiO and SiO 2 (see Sect. 2.4 and Table E.1).
All calculations start from chemical equilibrium abundances calculated using GGchem (Woitke & Helling 2021) for a solarlike composition (Asplund et al. 2009).A list of the considered species for the equilibrium calculation is given in Appendix B.

Nucleation
Nucleation reaction networks are ideally constructed by considering multiple reaction pathways.Unfortunately, only few such studies exist (see e.g.Bromley et al. 2016;Gobrecht et al. 2022;Andersson et al. 2023).In the kinetic network approach, the change in cluster number densities can be described as: where n N [cm −3 ] is the number density of a given polymer of size N (also called N-mer), F N is the set of forward reactions involving the N-mer, D N is the set of backward reactions involving the N-mer.
is the backward reaction rate coefficient of reaction r and J r is the number of reactants.
We describe the growth reactions of nucleating species as two-body reactions (a + b → c).The forward reaction rate coefficient k + r can then be described as (Peters 2017;Boulangier et al. 2019): where v r [cm s −1 ] is the relative velocity of the collision partners, α r (v r ) the sticking coefficient, σ r (v r ) [cm 2 ] is the reaction cross section and f (v r ) the relative velocity distribution of the colliding particles.Similar to other work, we set the sticking coefficient α j (ν r ) to 1 because detailed values for nucleation reactions are not available so far (e.g.Lazzati 2008;Bromley et al. 2016;Boulangier et al. 2019).The cross section is approximated by a collision of two hard spheres: where r 1 and r 2 [cm] are the interaction radii of the reaction partners.For this work, we consider radii including electrostatic forces (Köhn et al. 2021;Gobrecht et al. 2022;Kiefer et al. 2023).The relative velocity distribution is described by a Maxwell-Boltzmann distribution: where is the reduced mass [g] with m 1 , m 2 [g] being the masses of the reaction partners.Solving the integral from Eq. 3 yields: The backwards reaction rate (c → a + b) is derived by assuming detailed balance.For this, an equilibrium state needs to be defined which we assume to be the chemical equilibrium state such that the law of mass action can be applied: where p ⊖ = 10 5 Pa is the standard pressure and G ⊖ i (T gas ) [erg] is the Gibbs free energy of an i-mer at standard pressure.
For this study, we consider TiO 2 as nucleating species.The chemical network is extended by including all forward (Eq. 3) and backward reactions (Eq.8) from the monomer up to the 15-mer.The Gibbs free energy data is taken from Sindel et al. (2022).

Three-body reactions for cluster formation
Three-body reactions are important for the formation of small clusters.On the one hand, third bodies can remove the energy of formation from an association (forward) reaction thus increasing cluster formation rates.On the other hand, collisions with third bodies can induce dissociation (backward) reactions.For this work, we consider three-body reactions for the cluster formation of TiO 2 up to (TiO 2 ) 4 .The reaction rates are taken from Kiefer et al. (2023) (see reactions number RNr 19 to 26 in Table E.1).
To analyse for which temperature and pressures three-body reactions dominate over two-body reactions, we compare the reaction rate coefficients.To allow a direct comparison, we multiply the reaction rate coefficients with the number density of third bodies.The comparison can be seen in Fig. 1 reactions have the same pressure dependence, the scaling is the same for all of them.For two-body reactions, we use the following shorthand notation: For the association and dissociation of (TiO 2 ) 2 from and into two monomers respectively (RNr 19/20 and NU(1, 1)), the threebody reaction becomes dominant below T gas = 400 K and for pressures higher than p gas = 0.001 bar.Above T gas = 1300 K and for pressures lower than p gas = 10 bar, the two-body reaction becomes dominant.In between, either type of reaction can be dominant.At higher pressures (e.g.p gas > 1000 bar), the threebody reaction becomes dominant even at high temperatures (e.g.T gas > 2000 K).For all other reactions compared in this section (RNr 21/22, 23/24, 25/26 and NU(2, 1), NU(2, 2), and NU(3, 1)), the two-body reaction typically dominates above T gas > 400 K. Three-body reactions only start to become important at very high pressures (e.g.p gas > 1000 bar).

Bulk growth
In order to describe the bulk growth by gas-surface reaction we apply the moment method (Gail & Sedlmayr 1986, 1988;Dominik et al. 1993;Helling et al. 2001;Helling & Woitke 2006).The j-th moment L j [cm J g −1 ] is defined as: where j ∈ {0, 1, 2, 3}, V [cm 3 ] is the cloud particle volume, V l [cm 3 ] is the minimum volume of a cloud particle to start bulk growth, ρ [cm −3 ] the gas density and f (V) [cm −6 ] is the cloud Article number, page 3 of 21 particle size distribution function.Using these moments the following cloud particle properties can be derived (Gail & Sedlmayr 1988;Helling et al. 2001): where n d [cm −3 ] is the cloud particle number density, ⟨a⟩ [cm] the mean cloud particle radius, ⟨A⟩ [cm 2 ] the mean cloud particle surface area and ⟨V⟩ [cm 3 ] the mean cloud particle volume.
The change in the moments is determined by the nucleation and bulk growth and is described by the following set of equations 3 (Helling & Woitke 2006): where [cm −3 ] is the number density of the key gas-phase species, ν key r is the stochiometirc coefficient of the key gas-phase species for the surface reaction r, ⟨v r ⟩ [cm s −1 ] is the average relative velocity between the cloud particle and the key gas-phase species, S r is the reaction supersaturation (see section 2.5) and b surf r the surface area fraction of the given bulk growth material.χ net [cm s −1 ] is the net growth velocity.The key gas-phase species is the least abundant species involved in a given bulk growth reaction (Woitke & Helling 2003;Helling & Woitke 2006).The left term in the bracket of Eq. 17 represents the growth and the right term represents the evaporation.The cloud particle grows if the net sign of the bracket is positive and evaporates if the sign is negative.Similar to Helling & Woitke (2006), we assume that the surface area fraction can be approximated by the volume fraction: where A tot [cm 2 ] is the total cloud particle surface area and V tot [cm 3 ] is the total cloud particle volume.The volume fraction of each cloud particle material is tracked separately.The growth of cloud particles through bulk growth depletes the gas-phase.Therefore, we adjust the number densities n i of the species i involved in the surface reaction r (Helling & Woitke 2006): where δ(i) is equal to 1 for products and -1 for reactants.
3 Gravitational settling and other transport processes may be added as source terms to the r.h.s., (Woitke & Helling 2003) For this study, we consider

Reaction Supersaturation
To calculate the net growth velocity as described in Eq. 17, the reaction supersaturation needs to be calculated.Surface growth and evaporation of a material s can occur via three types of reactions: -Type 1: where X is a reactant, Y a product, A(N) a cloud particle containing N units of bulk growth material4 (e.g.Mg 2 SiO 4 would be 1 unit for Mg 2 SiO 4 [s]), F the set of reactants, D the set of products and ν i,r the stochiometric coefficients of species i for the surface reaction r.Type 1 reactions describe condensation, Type 2 reactions are chemical growth reactions, and Type 3 reactions involve surface chemistry.Cloud particles are assumed to be large enough for the following approximation to hold: where G ⊖ A(N) [erg] is the Gibbs free energy of formation at standard pressure of the A(N) cloud particle made from N units and G ⊖ A [erg] is the Gibbs free energy of formation of a solid unit at standard pressure.The goal of this section is to find the reaction supersaturation S r of these reactions defined as: where R f [s −1 ] is the growth rate and R b [s −1 ] is the evaporation rate.It is important to note that the following derivations are done in chemical disequilibrium.At no point in this section do we assume chemical equilibrium.

S r of Type 1 reactions (condensation)
Type 1 reactions are reactions where the reactant is also the bulk growth material.In this case the reaction supersaturation is equal to the supersaturation of the growth species: where p vap X [dyn cm −2 ] is the vapour pressure of species X, p X [dyn cm −2 ] is the partial pressure of species X, n X [cm −3 ] the gas-phase number density of species X, T gas [K] the temperature, and G ⊖ X [erg] the Gibbs free energy of formation of the condensing species in the gas-phase at standard pressure.Since the bulk growth material exists in the gas-phase, the supersaturation is well defined.

S r of Type 2 reactions
Type 2 reactions were discussed in detail in previous studies (Gail & Sedlmayr 1988;Gauger et al. 1990;Dominik et al. 1993;Patzer et al. 1998).The reaction supersaturation is given by (adapted from Patzer et al. 1998): where G ⊖ X [erg] and G ⊖ Y [erg] are the Gibbs free energies of formation of the gas-phase reactant and product, respectively.n X [cm −3 ] is the number density of the reactant X and n Y [cm −3 ] is the number density of the product Y.Type 2 reactions are well defined even if the bulk growth material is not present in the gasphase (see Patzer et al. (1998) for further details).

S r of Type 3 reactions
Type 3 reactions involve surface chemistry.Considering these reactions is especially important if the bulk growth material is not present in the gas-phase.For example, Mg 2 SiO 4 [s] can condense via the surface reaction To simplify the notation, we use the reaction of Eq. 27 as an example for this section and generalise the results in the end.
For our example in this section, we assume Mg to be the key gas-phase species.The bulk growth rate of this reaction can then be described by (adapted from Helling & Woitke 2006): where A A(N−1) [cm 2 ] is the surface area of the A(N − 1) cloud particle, n A(N−1) [cm −3 ] the number density of the A(N − 1) cloud particle, ν key [cm s −1 ] the relative velocity of the cloud particle and the key gas-phase species (e.g.Mg), and n key [cm −3 ] the number density of the key gas-phase species (e.g.Mg).Phase equilibrium for a given bulk growth reaction (short PGR, noted with • ) is characterised5 by the evaporation rate equalling the growth rate (R . Therefore according to Eq. 28, the evaporation rate is: where n • key [cm −3 ] is the number density of the key gas-phase species in PGR (e.g.Mg).Since all non-key gas-phase species are typically much more abundant then the key gas-phase species, their number densities in non-PGR only slightly differ to their number densities in PGR.Therefore, the following approximation holds: This allows us to write the reaction supersaturation as: leaving only the ratios of gas number densities in PGR to find.We start the derivation with a thought experiment.Imagine a box containing a given elemental abundance of Mg, SiO, H 2 O, H 2 , A(N − 1) and A(N).We assume that in this box, gas-phase species only react with each other via the specific surface reaction from Eq. 27 but do not react with each other otherwise.Over time, the box will evolve towards PGR for this specific chemical configuration.In PGR, the entropy of the box will be maximised which is equivalent to minimising the Gibbs free energy for the reactants and products of the given reaction.This minimisation problem with its constraints can be expressed in the following Lagrangian function: where E = {Mg, SiO, H 2 O, H 2 , A(N − 1), A(N − 1)} is the set of particles, N j the total number of particles j, N is the total number of gas particles, G ⊖ j [erg] the Gibbs free energy of formation of particle j at standard pressure, λ i [erg] are the Lagrangian multipliers and C i are constants.The constrains from λ 1 , λ 2 , and λ 3 are keeping the ratio of Mg, SiO, H 2 O and A(N − 1) per reaction constant using Mg as reference.The constraints from λ 4 , and λ 5 ensure mass conservation.Minimizing this Lagrangian for all molecules and cloud particles results in the following set of equations: In PGR, the Lagrangian function is minimal and thus the derivatives are zero.Solving this set of equations, using the approximation of Eq. 23, and going from particle numbers N j to particle Article number, page 5 of 21 A&A proofs: manuscript no.version_arxive number densities n j leads to: This result gives us the relation between the number densities of the reactants and products of the bulk growth reaction.Hence, the reaction supersaturation for the reaction of Eq. 27 is given by: For an arbitrary type 3 reaction that is limited by a key gas-phase species, the reaction supersaturation is then given by: where l X is the number of reactants and l Y is the number of products.If only 1 reactant is considered, this result matches the result for type 1 reactions.It also matches type 2 reactions if only 1 key reactant and 1 key product are considered.In the case of chemical equilibrium, our description of the supersaturation ratio for type 3 reactions is the same as the one found by Helling & Woitke (2006).
We define the right hand side of Eq. 40 as the reaction vapor coefficient c vap r [cm −3(l x −l y ) ] and fit it with: The fitting parameters for the surface reactions considered in this paper are given in Table E.2.This allows us to write the reaction supersaturation as: (44)

Formation rate of CCNs
To describe nucleation kinetically, we require the properties of each considered cluster size up until the size N ⋆ where the clusters become preferably thermally stable.For this, the Gibbs free energies G N (Eq.8), the interaction radii r N (Eq. 4 and 7) and the masses m N (Eq.6) of all considered cluster sizes need to be known.
which describes the rate at which CCNs are formed.The growth from clusters made from N max monomers to clusters with a volume of V l depletes the gas-phase of the clustering species.Because polymer nucleation is considered, all N-mers up to a given Q-mer are depleted: where n N [cm −3 ] are N-mer number densities and 1 ≤ N ≤ Q.
For TiO 2 , the change in the number density of clusters of size N max = 15 defines nucleation (see Eq. 45).We therefore exclude this size in the depletion description and select Q TiO 2 = 14.
Starting from V l , the particles can grow via surface growth.Similarly, they can shrink via evaporation down to size V l .To numerically separate between evaporation and nucleation, cloud particles should only evaporate down to size V l .For the ODE solver, we need a continuous transition and therefore adjust the evaporating surface area by multiplying the evaporation term of Eq. 17 with Including Eq.48 in Eq. 17 results in: For the rest of this paper, we are using Eq.49 for χ net in Eqs.16 and 19.

Exploring kinetic chemistry and nucleation
In Sect.3.1, we create a chemical network for the kinetic cloud formation model.Using this network, we study the impact on the gas phase when combining different chemical kinetics networks, nucleation and bulk growth.Describing bulk growth fully time dependent revealed a SiO-SiO 2 cycle within the surface reactions which is discussed in Sect.3.2.In Sect.3.3, we evaluate the nucleation rate's dependence on different maximum cluster sizes N max .

A chemical network for kinetic cloud formation
To find the impact of combining different chemical networks, nucleation and bulk growth on the number densities of the gasphase species, we compare the gas-phase abundances in 4 cases: -Equilibrium: Chemical equilibrium number densities calculated using GGchem.This calculation is time independent.
-VULCAN: The "NCHO thermo network" reactions of VULCAN.-Full: This network combines the "NCHO thermo network" reactions of VULCAN, the Ti and Si reactions as listed in Table E.1, the polymer nucleation reactions for TiO 2 as described in Sect.2.2, the formation of TiO 2 CCNs as described in Sect.2.6, and the bulk growth through condensation and surface reactions as described in Sect.2.4.
All simulations start from equilibrium number densities calculated with GGchem.All time axis in this paper measure the evaluation time t [s] of the disequilibrium chemistry starting from equilibrium conditions at t = 0 s.The results for the T gas -p gas point6 p gas = 0.002 bar at T gas = 1378 K can be seen in Fig. 2. The chemical equilibrium number densities match the predicted number densities of VULCAN and are therefore not shown in the figure .10 3 10 1 10 1 10 3 10 5 10 7 10 9 To compare the difference in the predicted number densities, we compare the maximum absolute difference between the logarithm of the number densities of species A between two chemical networks C 1 and C 2 : The P values for p gas = 0.002 bar and T gas = 1378 K, which represents a low-pressure level in a relatively cool atmosphere, are shown in Table 1.For the comparison we chose H 2 because it is the dominant gas phase species, H 2 O and CO 2 because they are commonly studied, methane (CH 4 ) because it has distinct spectral features, Mg and SiO 2 because they are condensing species, and TiO 2 because it is the nucleating species.
Comparing Equilibrium to VULCAN and VULCAN to VULCAN+ shows in both cases close to no difference (P < 0.1) in the predicted number densities of H 2 , H 2 O, CO 2 , CH 4 , and SiO 2 .In VULCAN+poly TiO 2 nucleation reactions and the formation of TiO 2 CCNs are added.Therefore, it comes at no surprise that the number density of TiO 2 decreases by close to two orders of magnitude.The impact for H 2 , H 2 O, CO 2 , CH 4 , and SiO 2 on the other hand is still negligible.When bulk growth is added in the full network, the number densities of TiO 2 , SiO 2 and Mg decrease by several orders of magnitude due to being depleted by the bulk growth processes.The impact of bulk growth can also be seen in the number densities of H 2 O, CO 2 , and CH 4 .The change in CH 4 is discussed in Sect.3.2.The number density of H 2 on the other hand is not significantly affected.
Also shown in Fig. 2 are the gas-phase concentrations calculated using GGchem including the equilibrium condensation of the bulk grow materials.Compared to our results, GGchem equilibrium condensation results predict higher gas-phase concentrations in the cloud forming species TiO 2 , SiO 2 , and Mg as well as the gas-phase only species H 2 O, CH 4 , and CO 2 .Because our work treats cloud formation kinetically, these differences can be caused by the nucleation or surface reactions which are both not considered within GGchem.In environments like the ISM or Titan's atmosphere, surface reactions are known to cause deviations from equilibrium gas-phase abundances (see Sect. 3.2).
Selecting a suitable gas-phase chemical kinetics network is important.We chose VULCAN because it includes commonly considered species such as H 2 O, CO 2 and CH 4 with a reasonably low number of reactions (780).Other chemical kinetics networks for exoplanet atmospheres include thousands of reactions (e.g.Venot et al. 2012;Venot & Agúndez 2015;Rimmer & Helling 2016;Hobbs et al. 2019;Venot et al. 2020).Because of the computational intensity of these networks, their evaluation is often limited to 1D models (e.g.Moses et al. 2005;Chadney et al. 2017;Hobbs et al. 2022;Barth et al. 2021).Adding kinetic nucleation and bulk growth to the chemical network can increase the computational time considerably.For the simulations in this section, the evaluation time doubled if nucleation and bulk growth were considered.If enough computational resources are available, our kinetic nucleation and bulk growth model can be combined with extensive chemical networks for a detailed study.Furthermore, models and observations have shown that the 3D structure of exoplanets can have an impact on the gas-phase chemistry (Baeyens et al. 2021;Prinoth et al. 2022;Lee et al. 2023).To evaluate the chemistry and cloud formation within 3D models, the cloud formation description can be combined with small but accurate networks (Tsai et al. 2022).
Most chemical kinetics networks for exoplanet atmospheres do not include many Mg, Ti, Si, or Fe bearing species.In our simulations, only the surface reactions c0, c1, c2, c4, c16, c71, c80, c85, c101, and c104 (see Table E.2) have all reactants and products within the chemical kinetics network.All other bulk growth reactions relay at least partially on gas-phase species which are only calculated in equilibrium.Ideally, all reactants and products of surface reactions should be included in the chemical kinetics network but they are not always available in literature and would drastically increase the number of reactions.The net process of this cycle is: This additional pathway for the dissociation of H 2 to H decreases the number density of CH 4 through the following reactions: Most of the carbon from CH 4 is transferred into H 2 CO with the following reaction: The change in carbon chemistry then also impacts other species like HCN, HCO, and C 2 H 2 .
To investigate at which pressures and temperatures the SiO-SiO 2 cycle significantly impacts the CH 4 abundance, P(CH 4 , Equilibrium, VULCAN + poly) values for a range of pressures and temperatures were calculated.The results are shown in Fig. 3.The difference in the CH 4 abundance is largest for pressures lower than p gas < 10 −3 bar and for temperatures around 1300 K.
The surface reactions used in this work were derived using a stoichiometric argument (Helling & Woitke 2006;Helling et al. 2008Helling et al. , 2017Helling et al. , 2019)).Unfortunately, more detailed studies of surface reactions of bulk growth materials are missing in literature.
To determine if all surface reactions of the SiO-SiO 2 cycle are likely to occur in exoplanet atmospheres, further investigations are needed.Because other processes, like quenching or photochemistry, can have similar effects (Moses et al. 2011), it will be difficult to gain insights into surface reactions through observations.Molaverdikhani et al. (2020) analysed the impact of clouds on the methane abundance.They found that clouds can increase the temperature in the photo-sphere which in turn reduces the methane abundance.In contrast to our work, they used condensation curves rather than surface reactions and therefore they did not observe a direct impact of cloud formation on the CH 4 abundance.
In the (ISM) surface reaction are discussed as sources for molecular hydrogen in the gas phase (Hollenbach & Salpeter 1971;Williams 2005;Sabri et al. 2013;Dishoeck 2014;Herbst 2014Herbst , 2017)).Similar to our study, in the ISM the surface of dust grains act as a catalyst but in contrast to our work, they do not result in bulk material being deposited.The calculation of ISM surface reaction rates typically accounts for the vibrational frequency of the reactants and the energy barriers between different sites on the dust particle (Dishoeck 2014).Our surface reaction description could be improved by similar considerations.However, the large number of surface reactions considered and the heterogeneity of the cloud particle make such evaluations difficult.
Similar to the ISM, the surfaces of aerosols in Titan's atmosphere can enhance the recombination of H into H 2 (Courtin et al. 1991;Bakes et al. 2003;Sekine et al. 2008).In addition, gas-phase catalytic cycles using hydrocarbons for the hydrogen recombination are postulated (Yung et al. 1984;Toublanc et al. 1995;Lebonnois et al. 2003).Both effects change the atomic hydrogen abundance which, similar to our work, can affect the abundance of hydrocarbons in return.In contrast to our work, the catalytic cycles considered are gas-phase only and do not result in bulk material being deposited.

Kinetic nucleation of TiO 2
To accurately model the CCN formation in exoplanet atmosphere, the nucleation rates of the dominant nucleating species need to be known.Previous studies determined TiO 2 (Goumans & Bromley 2013;Lee et al. 2015;Boulangier et al. 2019;Köhn et al. 2021)  which are discussed as nucleating species are Al 2 O 3 (Gobrecht et al. 2022), SiO (Gail & Sedlmayr 1986;Lee et al. 2015) and VO (Lecoq-Molinos et al. in prep).In addition to TiO 2 , we also analysed Al 2 O 3 as a possible nucleating species for clusters up to size N = 10 ( Gobrecht et al. 2022).The results were inconclusive (see Appendix C) and therefore, for this study, we decided to focus on TiO 2 .If the maximum cluster size is larger than the smallest thermally stable cluster (N max > N ⋆ ), the nucleation rate J ⋆ (V l ) is expected to be independent of the choice of the maximum clus-ter size.If the maximum cluster size is smaller than N ⋆ , we expect to see different nucleation rates depending on the choice of N max .Therefore, we test different N max (7 ≤ N max ≤ 15) and their impact on the cloud particle number density and nucleation rate.For this section, we set p gas = 0.02 bar and T gas = 1379 K.We use the full network as described in Sect.3.1.The cloud particle number densities and nucleation rates for different N max can be seen in Fig. 4. The predicted number densities for 7 ≤ N max ≤ 15 are all within a factor of 2. Furthermore, the peak in nucleation rate becomes smaller and appears later in time for larger clusters.
The biggest deviation in predicted cloud particle number density can be seen for N max = 13 and N max = 15 which predict lower cloud particle number densities than the rest.Looking at the Gibbs free energy per monomer unit (G ⊖ TiO 2 (N) /N) reveals that these sizes are the only N-mers that have a higher Gibbs free energy per monomer than their (N-1)-mers: Therefore, N = 13 and N = 15 are less thermally stable than their predecessors.Previous studies have shown the same preference for even N clusters (Lasserus et al. 2019) that we find for the (TiO 2 ) N clusters but further studies of larger sizes clusters are needed to determine if it is a size dependent trend that can affect the nucleation process.Because our nucleation rate is determined by the largest cluster size, having N max = 13 or N max = 15 naturally results in lower cloud particle number densities.Thermodynamic data for TiO 2 clusters larger than N = 15 is needed to further test this, and to find the thermally stable cluster size N ⋆ for TiO 2 .Few studies have already evaluated nucleation using a nonclassical approach.Lee et al. (2015) compared the nucleation rate predicted by CNT, MCNT and non-classical nucleation for various temperatures without considering surface growth.We calculated the cloud particle number density over the same temperature range as they analysed (Fig. 5).In contrast to our work, they considered only N max = 10 and only monomer nucleation.Both their and our study predict significant nucleation of TiO 2 for temperatures up to roughly 1300 K. Above that, the nucleation of TiO 2 quickly decreases.For temperatures between roughly 600 K to 1200 K, our model predicts approximately constant cloud particle number densities whereas the nucleation rate of Lee et al. (2015) decreases.This difference can be traced back to the polymer nucleation.Boulangier et al. (2019) showed that for TiO 2 and other nucleation seeds monomer nucleation can underestimate the formation of larger clusters in colder environments.

Time evolution of cloud formation in HD 209458 b
We simulate the chemistry and cloud formation for various T gasp gas points within HD 209458 b.The T gas -p gas profiles used in this paper were calculated using expert/MITgcm simulations of HD 209458 b conducted by Schneider et al. (2022).The T gasp gas profiles for the sub-stellar point, anti-stellar point, evening terminator, and morning terminator can be seen in Fig. 6.The cloud particle concentrations and the mean cloud particle size as well as selected gas-phase concentrations and volume fractions7 for the sub-stellar point, evening terminator, anti-stellar  (2022).The T gas -p gas points chosen for our simulations are marked with ⋆.
point and morning terminator at p gas = 0.002 bar can be seen in Fig. 7.This pressure layer was selected as it showed the largest spread in temperatures.The results for 4 logarithmically spaced pressure points (p gas ∈ {0.002 bar, 0.02 bar, 0.2 bar, 2 bar}) along the evening terminator can be seen in Fig. 8.Only one T gas -p gas profile was selected since the temperatures deeper in the atmosphere (p gas = 0.2 bar and p gas = 2 bar) are similar.The evening terminator was selected because of its intermediate temperature in the upper atmosphere at p = 0.002 bar.We use the full network as described in Sect.3.1 for all simulations and start from chemical equilibrium calculated with GGchem.
To be able to compare gas-phase timescales to cloud formation timescales, we ran the VULCAN+ network (see Sect. 3.1) starting from solar-like atomic abundances for the T gas -p gas points of the evening terminator.The resulting gas-phase concentrations of selected gas-phase species can be seen in Fig. 9.For all T gas -p gas points, the concentrations of H 2 O, TiO 2 , and SiO 2 quickly approach their stationary values (τ chem < 1 s).Their chemical timescale is highly pressure dependent and decreases for higher pressures.CH 4 and NH 3 on the other hand, still show significant differences for t > 10 5 s for all but the highest pressure (p gas = 2 bar).

Time evolution of cloud formation
Our results for the cloud formation within HD 208458 b start from a chemical equilibrium gas-phase from which clouds are formed.These simulations therefore can give us an indication on the timescale of nucleation and bulk growth.The predicted cloud particle concentrations for most T gas -p gas points of HD 209458 b quickly converge to stationary values (τ nuc < 1 s).The only exception is the sub-stellar point at p gas = 0.002 bar where no cloud particles are predicted due to the high temperatures.Comparing the chemical timescales (see Fig. 9 or Tsai et al. 2018;Mendonça et al. 2018)   T gas -p gas points at p gas = 0.002 bar, other than the sub-stellar point, show only a small temperature dependence of the predicted cloud particle concentrations and nucleation timescale.This is not unexpected for temperatures ranging from 919 K to 1378 K.We have shown that TiO 2 nucleation is roughly constant for this temperature range (see Sect. 3.3).Similarly, along the evening terminator we see a decrease in cloud particle concentrations consistent with the findings of Sect.3.3.At p gas = 2 bar within the evening terminator, the cloud particle concentration reaches only n d /n gas ≈ 10 −22 .The lower number of cloud particles results in larger cloud particles because the bulk growth ma-  terial condense onto fewer particles (Helling et al. 2023).Hence, the average cloud particle size reaches up to 0.033 cm.The peak in bulk growth closely follows the peak in nucleation and also approaches stationary values on timescales shorter than 1 second (τ bulk < 1 s).The exception to this is the evening terminator at p gas = 2 bar.The cloud particles still grow at roughly the same speed, but since much more material is available per cloud particle, it takes longer to reach stationary values for the average cloud particle size.
In all cases, the volume fractions start out TiO 2 dominated.After bulk growth starts the cloud particles become considerably heterogeneous.In all cases, Mg 2 SiO 4 becomes the domi- nant bulk material and hence also the dominant Mg and Si bearing species.Around 0.1 to 1 second, a short increase in the average cloud particle size can be seen for the morning terminator and the anti-stellar point at p gas = 0.002 bar.This size increase is caused by a temporary increase in SiO[s], SiO 2 [s] and Fe 2 SiO 4 [s] (see also Fig. F.1). Without the SiO-SiO 2 cycle the temporary peak of Fe 2 SiO 4 still occurs.This temporary increase is likely a result of feedback between cloud formation and disequilibrium chemistry.Because the cloud formation is directly coupled to the gas phase via the reaction supersaturation ratio, temporary changes in the gas-phase chemistry can be caused by cloud formation and vice versa.
For the dominant Fe bearing species we see a switch from Fe 2 SiO 4 to Fe 2 O 3 for t > 10 3 s.These are similar timescales of CH 4 and NH 3 (see Fig. 9 or Tsai et al. 2018;Mendonça et al. 2018).It is important to note that the change in the dominant Fe bearing species is not related to the SiO-SiO 2 cycle.It still occurs even if SiO[s] and SiO 2 [s] are not considered as bulk growth species.The timescale of the transition is highly pressure dependent and becomes shorter for higher pressures.Furthermore, the change in composition does not significantly affect the cloud particle size.Here it is important to note that our cloud formation formalism does not include any solid-to-solid composition changes.All changes happen via the gas-phase through bulk growth reactions as described in Sect.2.5.Helling & Woitke (2006) analysed the timescales of cloud formation with a similar cloud model as used in this work.In contrast to our work, the gas-phase is assumed to be in equilibrium (and depleted by cloud formation), MCNT is used to Article number, page 11 of 21 describe nucleation, and fewer surface reactions are used.The nucleation and bulk growth timescales they find are similar to ours (τ nuc < 1 s and τ bulk < 1 s).Coupling gas-phase chemistry and cloud formation does not seem to impact these timescales.However, secondary effects like the change in the dominant Febearing species and the SiO-SiO 2 cycle only appear once gasphase chemistry and cloud formation are fully coupled.
Powell et al. ( 2018) also analysed the timescales of cloud formation using a diffusive approach.Their rates are limited by the time it takes for the key species to diffuse to the cloud particle.They calculate their timescales as the number density of cloud particles divided by the influx of new cloud particles once a stationary solution has been reached.Consequently, their growth and nucleation timescales for TiO 2 are larger than ours (τ nuc > 10 s and τ bulk > 10 s).

Comparison to dynamical processes
To find whether cloud formation happens in disequilibrium or is affected by disequilibrium chemistry, we compare our results to different dynamical processes.

Gravitational settling
Cloud particles in exoplanet atmospheres gravitationally settle over time.Whether gravitational settling timescales are faster than cloud formation timescales depends on many factors such as the bulk growth speed, bulk growth material replenishment, and the frictional force of cloud particles within the atmosphere (Woitke & Helling 2003).For smaller particles (⟨a⟩ < 10 −4 cm), growth is generally more efficient than gravitational settling (Woitke & Helling 2003;Powell et al. 2018).However, if the condition favour larger particles and gravitational settling becomes more efficient "cold traps" can occur (Parmentier et al. 2013(Parmentier et al. , 2016;;Powell et al. 2018) where most cloud material is concentrated at the cloud base.
Comparing our cloud particle number densities and average radii to Powell et al. (2018) reveals differences that can be explained by gravitational settling and replenishment.In contrast to our work, they generally predict less and larger cloud particles.Settling removes cloud particles from the atmosphere thus leading to less particles.The replenished material then condenses onto already existing particles.Since fewer particles are present, they become larger.The exception to this is the evening terminator at at p gas = 2 bar.Here, nucleation is so inefficient that we predict very large cloud particles (⟨a⟩ = 0.033 cm).However, these particles would quickly settle down and they are unlikely to persist in a 1D model.

Vertical and horizontal transport
Quenching occurs when the chemical timescale τ chem [s] becomes larger then the vertical mixing timescale τ dyn [s] (Moses 2014).In low density environments, chemical timescales typically become longer (Tsai et al. 2018) and dynamical timescales typically become smaller (Parmentier et al. 2013).Therefore, quenching becomes more relevant in the upper atmosphere of exoplanets (Baeyens et al. 2021).Typical vertical mixing timescales are between 10 3 s < τ dyn < 10 7 s (Agundez et al. 2014;Drummond et al. 2018;Baeyens et al. 2021).Our results show that nucleation happens on much shorter timescales than this (τ nuc < 1 s) and is therefore expected to be less affected by quenching.The cloud particle composition on the other hand changes on longer timescales which are similar to the chemical timescales of CH 4 and NH 3 .Both CH 4 and NH 3 are known to be gas-phase species affected by quenching (Moses et al. 2011).Therefore, the cloud particle composition might be susceptible to quenching as well.
Similar to the vertical timescale, one can compare the nucleation and bulk growth timescales to the horizontal mixing timescales which consists of the latitudinal timescale and the longitudinal timescale.Mendonça et al. (2018) analysed8 WASP-43 b and found that latitudinal mixing happens on similar timescales as the vertical timescale.Longitudinal mixing on the other hand can be orders of magnitude shorter.This is mostly due to the strong equatorial wind jets.In their analysis, all (longitudinal, latitudinal and vertical) mixing timescales are well above 10 3 s > τ dyn .Therefore, similar to quenching, nucleation and the peak in bulk growth might be less affected by horizontal mixing.The cloud particle composition on the other hand might be impacted.

Stellar Flares
If periodic effects disturb the chemical abundances, the relaxation timescale τ relax indicates how quickly the chemical abundances return back to pre-disruption values.An example for an effect that temporarily impacts chemistry are stellar flares.They periodically enhance the radiation received by a planet and cause chemical disequilibrium through photochemistry (Tilley et al. 2019;Louca et al. 2022).The chemical relaxation timescale after a stellar flare event can be on the order of hours (τ relax > 10 3 s; Konings et al. 2022).In the chemical relaxation scheme (Smith 1998;Cooper & Showman 2006;Kawashima & Min 2021) the relaxation timescale is given by the chemical timescale.This scheme typically also finds relaxation timescales of τ relax > 10 3 s for CH 4 and NH 3 (see Fig. 9 or Tsai et al. 2018;Mendonça et al. 2018).Nucleation and the peak in bulk growth occur on much shorter timescales than this and therefore can adjust to the temporary chemical disequilibrium.The cloud particle composition on the other hand takes longer to transition and might not adjust to the temporary chemical disequilibrium.

Summary
We established a fully kinetic cloud formation description combining disequilibrium chemistry, kinetic nucleation and bulk growth through condensation and surface reactions.The kinetic gas-phase chemistry network for this study was based on the "NCHO thermo network" of VULCAN.This network was expanded with Ti (and Si) reactions to connect the gas-phase chemistry with the kinetic nucleation of TiO 2 .We considered TiO 2 polymer nucleation using cluster data up to cluster size N = 15.For the bulk material, we considered TiO 2 and Fe 2 SiO 4 [s].These materials can grow through 59 bulk growth reactions.
The kinetic polymer nucleation of TiO 2 indicates a similar temperature and pressure dependence as previous non-classical studies.We tested different maximum cluster sizes between N max = 7 and N max = 15 and found differences of the predicted cloud particle number density within a factor of 2. For further investigations, thermodynamic data of larger TiO 2 clusters is required.Overall, our results suggest that kinetic nucleation is a viable alternative to classical nucleation theory if the cluster data of the nucleating species is available.The required cluster sizes depend on the nucleating species.
The fully kinetic description of surface reactions resulted in a SiO-SiO 2 catalytic cycle that dissociates H 2 into 2 H.The increase in atomic hydrogen can reduce the CH 4 abundance by over an order of magnitude.If this catalytic cycle occurs in exoplanet atmospheres remains to be seen.
We simulated the chemistry of various T gas -p gas points within the atmosphere of HD 209458 b.For all T gas -p gas points, except the sub-stellar point, nucleation and bulk growth occurred.In all cases were nucleation occurred, nucleation and bulk growth reached a stationary behavior within 1 s.A comparison to the timescales of quenching and chemical relaxation showed that nucleation can happen on much shorter timescales.Hence, our work confirms that the assumption of localised nucleation is generally justified in exoplanet atmospheres.For the cloud particle composition on the other hand, we found changes on the timescale larger than 10 3 seconds.This indicates that the composition of cloud particles can be susceptible to quenching.reactions are derived using detailed balance with the following coefficients: The condensation reactions taken from

Fig. 3 .
Fig. 3. Differences in the CH 4 abundance between chemical equilibrium and the VULCAN+poly network for various temperatures and pressures.

Fig. 4 .Fig. 5 .
Fig.4.Cloud particle number densities (Top) and nucleation rates (Bottom) for TiO 2 nucleation with different N max at p gas = 0.02 bar and T gas = 1379 K.

Fig. 6 .
Fig. 6.T gas -p gas profiles of HD 209458 b taken from Schneider et al. (2022).The T gas -p gas points chosen for our simulations are marked with ⋆.

Fig. 7 .
Fig. 7. Concentrations of selected gas-phase species (Top), cloud particle number density (Upper middle), mean cloud particle size (Lower middle), and selected volume fractions (Bottom) at the sub-stellar point, evening terminator, anti-stellar point and morning terminator at p gas = 0.002 bar.The sub-stellar point does not form clouds.

Fig. 8 .
Fig. 8. Concentrations of selected gas-phase species (Top), cloud particle number density (Upper middle), mean cloud particle size (Lower middle), and selected volume fractions (Bottom) for logarithmically spaced pressures along the evening terminator.

Fig. 9 .
Fig.9.Concentrations of selected gas-phase species along the evening terminator starting from solar-like atomic abundances.
Fig. D.2.Comparison of the chemical network of Gobrecht and Vulcan.
is the nucleation rate (see Sect. 2.6), C d is the set of surface reactions, ∆V r [cm 3 ] is the volume increase per surface reaction r, n SiO 4 [s] as bulk growth materials and include the surface reactions listed in Table E.2.
This network combines the "NCHO thermo network" reactions of VULCAN, and the Ti and Si reactions as listed in TableE.1-VULCAN+poly:Thisnetworkcombines the "NCHO thermo network" reactions of VULCAN, the Ti and Si reactions as listed in TableE.1, the polymer nucleation reactions for TiO 2 as described in Sect.2.2, and the formation of TiO 2 CCNs as described in Sect.2.6.
Article number, page 6 of 21 S.Kiefer et al.:Fully time-dependent cloud formation from a non-equilibrium gas-phase in exoplanetary atmospheres -VULCAN+: Concentrations of selected gas-phase species for p gas = 0.002 bar at T gas = 1378 K using different chemical kinetics networks.The diamond shaped marker show GGchem results including equilibrium condensation.

Table 1 .
Offset values between the number densities of given molecular species for different chemical networks according to Eq. 50.
to the nucleation timescale shows that nucleation happens on similar timescale as the chemical species with a shorter chemical timescale (e.g.H 2 O, TiO 2 , SiO 2 ).The Kiefer et al.:Fully time-dependent cloud formation from a non-equilibrium gas-phase in exoplanetary atmospheres Article number, page 10 of 21 S.