Exact phase structure of AdS black holes surrounded by quintessence dark energy

Treating the cosmological constant as a thermodynamic pressure, charged black holes always exhibit the small/large black-holes phase transition in the extended thermodynamic phase space, while their complicated equation of state could make it difficult to give an analytical and exact picture for the critical points and phase transition. In this paper, we focus on the number of physical critical points to consider the exact phase structure and the effect of charge. An exact bound of charge 0 < Q < Q(c) restricting the small/large black-holes phase transition is presented, by considering RN-AdS black holes surrounded by quintessence dark energy. This bound results from the constraint of positive temperature condition. It is shown that only black holes with small charge contain the small/large black-holes phase transition, while black holes with big charge also have P-nu oscillatory behavior and the classical "swallow tail" occurring at un-physical negative temperature. This method can be generalized to the study on the exact phase structures of small/large black-holes phase transition and reentrant phase transition, for other complicated thermodynamic systems with charge and angular momentum. Copyright (C) EPLA, 2018