Casimir energy and entropy in the sphere-sphere geometry

We calculate the Casimir energy and entropy for two spheres described by the perfect-metal model, plasma model, and Drude model in the large-separation limit. We obtain nonmonotonic behavior of the Helmholtz free energy as a function of separation and temperature for the perfect-metal and plasma models, leading to parameter ranges with negative entropy, and also we obtain nonmonotonic behavior of the entropy as a funtion of temperature and the separation between the spheres. This nonmonotonic behavior has not been found for the Drude model. The appearance of this anomalous behavior of the entropy as well as its thermodynamic consequences are discussed.