Joule-Thomson Expansion of RN-AdS Black Holes in $f(R)$ gravity

In this paper, we study Joule-Thomson effects for charged AdS black holes in $f(R)$ gravity. We obtain the inversion temperatures as well as inversion curves, and investigate similarities and differences between van der Waals fluids and charged AdS black holes in $f(R)$ gravity for this expansion. In addition, we determine the position of the inversion point versus different values of the mass $M$, the charge $Q$ and the parameter $b$ for such black hole. At this point, the Joule-Thomson coefficient $\mu$ vanishes, an import feature that we used to obtain the cooling-heating regions by scrutinizing the sign of the $\mu$ quantity.

the thermodynamical system is a black hole, more precisely its mass. JT expansion [32] is a convenient isoenthalpic tool that a thermal system exhibits with a thermal expansion, where the Joule-Thomson coefficient µ JT = ∂T ∂P H is the main quantity to discriminate between the cooling and heating regimes of the system. It is worth noting that when expanding a thermal system with a temperature T , the pressure always decreases yielding a negative sign to ∂P . In this context, we can consider two different regimes with respect to the socalled the inversion temperature, defined as the temperature T i at which the Joule-Thomson coefficient vanishes µ JT (T i ) = 0 : If T < T i (T > T i ), then the Joule-Thomson processus cools (warms) the system with ∂T < 0 and µ JT > 0 (∂T > 0 and µ JT < 0) respectively. When the system temperature tends to T i , its pressure is referred as the inversion pressure P i , so defining a special point called the inversion point (T i , P i ) at which the cooling-heating transition occurs.
The outline of this work is as follows: In the next section, we review briefly the essential of the thermodynamic properties and stability of the charged-AdS black hole solution in f (R) background. In section 3, we study the JT expansion under constant mass, and derive the inversion temperature T i as well as the corresponding inversion point of such black hole. We also show when the cooling phase is changed to heating phase at a particular (inversion) pressure P i . The last section is devoted to our conclusion.

Thermodynamic of charged AdS black holes in f (R) gravity background
In this section, we briefly review the main features of the four-dimensional charged AdS black hole corresponding in the R + f (R) gravity background with a constant Ricci scalar curvature [26]. The action is given by Here, R denotes the Ricci scalar curvature while f (R) is an arbitrary function of R. In addition F µν stands for electromagnetic field tensor given by F µν = ∂ µ A ν − ∂ ν ∂ µ , where A µ is the electromagnetic potential. From the action (1), the equations of motion for gravitational field g µν and the gauge field A µ are, The analytic solution of equation (2) has been determined in [26] when the Ricci scalar curvature is constant R = R 0 = const, where in this simple case, (2) simplifies to: The charged static spherical black hole solution of (2) in in 4d gravity model takes the form [26] where the metric function N (r) is given by, while the two parameters m and q are proportional to the black hole mass and the charge respectively [33] In this background, the electric potential Φ can be evaluated as where the black hole event horizon r + denotes the largest root of the equation N (r + ) = 0. At the event horizon, one can also derive the Hawking temperature as well as the entropy of this kind of black hole [26,33], Recalling the analogy between the cosmological constant and the thermodynamics pressure, while its corresponding conjugate quantity is identified to the volume [9,34,35], one can deduce the following relations, and At this stage, it is straightforward to see that the above black hole quantities satisfy to the following Smarr relation: Furthermore, by taking into account the F (R) corrections with a constant Ricci scalar, the first law of thermodynamics is written as: At last, from the equations of the Hawking temperature (9) and the pressure (11) of such black hole, one can easily derive the corresponding equation of state, P = P (T, r + ), It is worth noting here that the F (R) background induced corrections to the pressure (11) and to the subsequent formulas can bring to light new possible feature which might be revealed through the phase transition structure of the charged-AdS black holes. Next section will be devoted to verify this proposal by means of Joule-Thomson expansion.

Joule-Thompson expansion of Charged black hole in f (R) gravity
Applying method similar to the one used in [36], we consider Joule-Thomson expansion for charged-AdS black holes in f (R) gravity. For a fixed charge the Joule-Thomson coefficient is given by, besides, the equation of the state of such black hole is provided in terms of thermodynamic volume by substituting (12) in (15), the equation (9) transforms to: by using Eq.(17) this into the right hand side of Eq.(16), one can derive the temperature corresponding to a vanishing Joule-Thomson coefficient, dubbed inversion temperature T i : T i can also be rewritten in terms of its corresponding pressure: By subtracting Eq. (18) form Eq. (19), we get the following polynomial equation, which possesses four roots. Here, we only consider the real positive root given by, Once this root is substituted into Eq. (19), the inversion temperature becomes, so when inversion pressure P i vanishes, the inversion temperature reaches its minimum at: Consequently, the critical temperature is just twice the value of the inversion temperature, in perfect agreement with the result of [29]. In Fig.1 we plot the inversion curves for charged AdS black hole for different values of charge Q. We can see that, in contrast to Van der Waals fluids, there is only a lower inversion curve which does not terminate at any point, since the expression inside the square root of Eq.(22) is always positive. This feature has also been observed in [29] with charged AdS black holes as well in [30] with the rotating AdS black holes.
Next, thanks to Eq.(7) and Eq.(15), we also illustrates in Fig.2 the isenthalpic curve corresponding to a constant mass in the (T, P )-diagram.
From Fig.2, we see that the inversion curves divide the space (T, P ) into two separated regions: The region above the inversion curves corresponds to the cooling region, while the region under the inversion curves corresponds to the heating one. Note that one can discriminate between the cooling / heating regions just by checking the sign of the isenthalpic curves slope. The sign of the slope is positive in the cooling region and it flip to negative in the heating region. The cooling / heating phenomena never takes place on the inversion curve which plays the role of a separating boundary between the two regions.

Conclusion
In this paper, we have studied the Joule-Thomson expansion for RN-AdS black hole in f (R) gravity background in the context of the extended phase space, where the cosmological constant is identified with the pressure. Here, the black hole mass is interpreted as an enthalpy, so we assume that mass does not change during the Joule-Thomson expansion. Our analysis has shown that the inversion curve always corresponds to the lower curve. This means that black hole always cools above the inversion curve during the expansion. At last, we have identified the cooling and heating regions for different values of the parameter b and the black hole mass M .