  In this paper, we study spontaneous scalarization of asymptotically anti-de
Sitter charged black holes in the Einstein-Maxwell-scalar model with a
non-minimal coupling between the scalar and Maxwell fields. In this model,
Reissner-Nordstr\"om-AdS (RNAdS) black holes are scalar-free black hole
solutions, and may induce scalarized black holes due to the presence of a
tachyonic instability of the scalar field near the event horizon. For RNAdS and
scalarized black hole solutions, we investigate the domain of existence,
perturbative stability against spherical perturbations and phase structure. In
a micro-canonical ensemble, scalarized solutions are always thermodynamically
preferred over RNAdS black holes. However, the system has much rich phase
structure and phase transitions in a canonical ensemble. In particular, we
report a RNAdS BH/scalarized BH/RNAdS BH reentrant phase transition, which is
composed of a zeroth-order phase transition and a second-order one.
