  Aims: The secondary atmospheres of terrestrial planets form and evolve as a
consequence of interaction with the interior over geological time. We aim to
quantify the influence of planetary bulk composition on the
interior--atmosphere evolution for Earth-sized terrestrial planets to aid in
the interpretation of future observations of terrestrial exoplanet atmospheres.
Methods: We used a geochemical model to determine the major-element composition
of planetary interiors (MgO, FeO, and SiO2) following the crystallization of a
magma ocean after planet formation, predicting a compositional profile of the
interior as an initial condition for our long-term thermal evolution model. Our
1D evolution model predicts the pressure-temperature structure of the interior,
which we used to evaluate near-surface melt production and subsequent volatile
outgassing. Volatiles are exchanged between the interior and atmosphere
according to mass conservation. Results: Based on stellar compositions reported
in the Hypatia catalog, we predict that about half of rocky exoplanets have a
mantle that convects as a single layer (whole-mantle convection), and the other
half exhibit double-layered convection due to the presence of a mid-mantle
compositional boundary. Double-layered convection is more likely for planets
with high bulk planetary Fe-content and low Mg/Si-ratio. We find that planets
with low Mg/Si-ratio tend to cool slowly because their mantle viscosity is
high. Accordingly, low-Mg/Si planets also tend to lose volatiles swiftly
through extensive melting. Moreover, the dynamic regime of the lithosphere
(plate tectonics vs.\ stagnant lid) has a first-order influence on the thermal
evolution and volatile cycling. These results suggest that the composition of
terrestrial exoplanetary atmospheres can provide information on the dynamic
regime of the lithosphere and the thermo-chemical evolution of the interior.
