Phase transition and thermodynamic geometry of f(R) AdS black holes in the grand canonical ensemble

The phase transition of a four-dimensional charged AdS black hole solution in the R + (R) gravity with constant curvature is investigated in the grand canonical ensemble, where we find novel characteristics quite different from that in the canonical ensemble. There exists no critical point for T - S curve while in former research critical point was found for both the T - S curve and T - r(+) curve when the electric charge of f(R) black holes is kept fixed. Moreover, we derive the explicit expression for the specific heat, the analog of volume expansion coefficient and isothermal compressibility coefficient when the electric potential of f(R) AdS black hole is fixed. The specific heat C-Phi encounters a divergence when 0 < Phi < b while there is no divergence for the case Phi > b. This finding also differs from the result in the canonical ensemble, where there may be two, one or no divergence points for the specific heat C-Q. To examine the phase structure newly found in the grand canonical ensemble, we appeal to the well-known thermodynamic geometry tools and derive the analytic expressions for both the Weinhold scalar curvature and Ruppeiner scalar curvature. It is shown that they diverge exactly where the specific heat C-Phi diverges.