Weinhold geometry and thermodynamics of Bardeen AdS black holes

Thermodynamics of Bardeen AdS black holes has attracted a great deal of attentions due to its intrinsic complications and rich phase structures. However, the entropy and thermodynamic volume are incorrect in some literatures. In this paper we revisit the thermodynamics of Bardeen AdS black holes and provide the correct entropy and thermodynamic volume. Furthermore, thermodynamic geometries are a powerful tool to probe the microstructure of black holes. Based on the Hessian matrix of black hole mass, we introduce a thermodynamic metric and give its scalar curvature in Weinhold's geometry. The conformal relation between Weinhold's geometry and Ruppeiner's geometry will be changed due to the specific first law of thermodynamics for regular black holes like the Bardeen AdS black hole. We also investigate the critical behaviour of phase transitions in an extended phase space, and find that the critical behaviour of the Bardeen AdS black hole coincides with that of liquid-gas systems. In particular, based on the Weinhold geometry we unveil a repulsive interaction in the microstructure of the Bardeen AdS black hole under its small volume state, while it is known that only attractive interaction exists in the microstructure of the van der Waals fluid.