Dust in brown dwarfs and extra-solar planets. VIII. TiO$_2$ seed formation: 3D Monte Carlo versus kinetic approach

Abstract: We aim to understand the onset of cloud formation {and study the formation of} TiO$2$-CCNs. The formation of (TiO$_2$)${\rm N}$ clusters as precursors to extrasolar cloud formation is modelled by two different methods in order to understand their potential, identify underlying shortcomings, and to validate our methods. We propose potential spectral tracers for TiO$2$-CCN formation. We applied three-dimensional Monte Carlo (3D MC) simulations to model the collision-induced growth of TiO$_2$-molecules to (TiO$_2$)${\rm N}$-clusters in the free molecular flow regime of an atmospheric gas. We derived individual, time-dependent (TiO$2$)${\rm N}$ cluster number densities. For $T=1000$K, the results are compared to a kinetic approach that utilises thermodynamic data for individual (TiO$2$)${\rm N}$ clusters. The {(TiO$2$)${\rm N}$} cluster size distribution is temperature dependent and evolves in time until a steady state is reached. For $T=1000$K, the 3D MC and the kinetic approach agree well regarding the cluster number densities for $N=1\,\ldots\,10$, the vivid onset of cluster formation, and the long transition into a steady state. Collision-induced growth and evaporation simulated using a 3D MC approach enables a faster onset of cluster growth through nucleation bursts. Different size distributions develop for monomer-cluster and for cluster-cluster growth, with the largest clusters appearing for cluster-cluster growth. The (TiO$_2$)$_N$ cluster growth efficiency has a sweet-spot temperature at $\approx 1000$K at which CCN formation is triggered. The combination of local thermodynamic conditions and chemical processes therefore determines CCN formation efficiency.