Bottom-up dust nucleation theory in oxygen-rich evolved stars I. Aluminium oxide clusters

Aluminum oxide (alumina, Al$_{2}$O$_{3}$) is a promising candidate as a primary dust condensate in the atmospheres of oxygen-rich evolved stars. Therefore, alumina \textit{seed} particles might trigger the onset of stellar dust formation and of stellar mass loss in the wind. However, the formation of alumina dust grains is not well understood.} {To shed light on the initial steps of cosmic dust formation (i.e. nucleation) in oxygen-rich environments by a quantum-chemical bottom-up approach.} {Starting with an elemental gas-phase composition, we construct a detailed chemical-kinetic network describing the formation and destruction of aluminium-bearing molecules and dust-forming (Al$_{2}$O$_{3}$)$_n$ clusters up to the size of dimers ($n$=2) coagulating to tetramers ($n=$4). Intermediary species include the prevalent gas-phase molecules AlO and AlOH, and Al$_x$O$_y$ clusters with $x=$1$-$5, $y=$1$-$6. The resulting extensive network is applied to two model stars, representing a semi-regular variable and a Mira-type, and to different circumstellar gas trajectories including a non-pulsating outflow and a pulsating model. The growth of larger-sized (Al$_{2}$O$_{3}$)$_n$ clusters with $n=$4$-$10 is described by the temperature-dependent Gibbs free energies of the most favourable structures (i.e. the global minima clusters) as derived from global optimisation techniques and calculated by density functional theory. We provide energies, bond characteristics, electrostatic properties and vibrational spectra of the clusters as a function of size $n$ and compare these to corundum corresponding to the crystalline bulk limit ($n\rightarrow \infty$). {The circumstellar aluminium gas-phase chemistry in oxygen-rich giants is primarily controlled by AlOH and AlO, which are tightly coupled by the reactions AlO+H$_2$, AlO+H$_2$O, and their reverse. Models of ...}