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Here we extend the approach developed in \cite{adel_2} to study thethermodynamics of Taub-NUT-AdS and dyonic Taub-NUT-AdS solutions. Furthermore,we investigate in details the possible phase structures of the dyonicTaub-NUT-AdS solution. We show that the first law, Gibbs-Duhem and Smarr'srelations are all satisfied for both solutions. Our study of phase structuresshows some intriguing features, which were not reported before, among which theexistence of two distinguished critical points with a region of continuousphase transitions in between, and the possibility of merging them into onepoint. To analyze these phases we consider both canonical and mixed ensembles.The two distinguished critical points occur for the canonical case as well asthe mixed cases with $1/2 \le \phi_e < 1$. Another interesting case is themixed ensemble with $\phi_e \ge 1$, where we have one critical point but thecontinuous phase transition region in the $P-T$ diagram is close to the origin,in contrast with what happens in Reissner-Nordstrom-AdS solutions and Van derWaals fluids, i.e., the continuous phase transition happens only for low enoughpressures and temperatures!

Dyonic Taub-NUT-AdS: Unconstraint Thermodynamics and Phase Structure