Thermodynamic Criticality and Phase Transitions in Dilaton Black Holes with Nonlinear Electrodynamics: A Geometrothermodynamics Study

We investigate the thermodynamic criticality and phase structure of dilaton black holes in the presence of Born-Infeld nonlinear electrodynamics and a Liouville-type potential. By adopting an analytical approach based on the near-horizon geometry and the first law of thermodynamics, we derive consistent expressions for temperature, entropy, charge, and mass. This method allows us to bypass the complexity of finding a global closed-form metric solution while maintaining exact thermodynamic consistency. In the extended phase space, where the cosmological constant is treated as thermodynamic pressure, we identify a first-order phase transition between small and large black hole phases, culminating in a second-order critical point. The critical exponents are computed analytically and found to belong to the mean-field universality class, with α = 0, consistent with standard Van der Waals systems. Using Ruppeiner geometry, we study the thermodynamic state space and show that the scalar curvature diverges negatively at the critical point, indicating that the underlying microscopic interactions are predominantly attractive. The dilaton coupling α enhances this attraction, suggesting that the scalar field plays an active role in the condensation mechanism. Our results highlight the significant impact of non-minimal scalar couplings and nonlinear electrodynamics on black hole thermodynamics.