Thermophysics of colloidal systems confined to spherical surfaces

A two-dimensional Brownian fluid confined, in equilibrium, to a closed surface is necessarily a finite system that consequently does not fit the benchmark of an extensive thermodynamic system (ETS). Considering the particular case of a spherical surface, and in the absence of external fields, one gets a uniform arrangement which in the thermodynamic limit obeys the Euler equation that defines an ETS: the Landau (grand) potential becomes identical to the negative of the surface pressure times the area. The best-known upshot from this condition is the proportionality of the isothermal (surface) compressibility to the particle number fluctuations. The present study implements a bespoke Brownian dynamics approach, complemented with the customary methods from statistical mechanics, to show the extent to which a system with a finite radius (still defined by its lack of boundaries) departs from these ETS criteria. An experimental realization of this kind of system can be found in Pickering emulsions, which are stabilized by charged surfactant nanoparticles adsorbed to each one of the spherical oil droplets immersed in water. It is thus shown, by applying a model pair potential suitable for the interactions among these adsorbates (whose intensity can be adjusted), that the deviation from the ETS reference grows rapidly as the repulsion among the nanoparticles increases. Moreover, it is also demonstrated that this has important implications for the closure relations employed for the determination of the pair-correlation functions within typical liquid theory approaches.