The orientational structure of a model patchy particle fluid: Simulations, integral equations, density functional theory, and machine learning

We investigate the orientational properties of a homogeneous and inhomogeneous tetrahedral four-patch fluid (Bol-Kern-Frenkel model). Using integral equations, either (i) HNC or (ii) a modified HNC scheme with a simulation input, the full orientational dependence of pair and direct correlation functions is determined. Density functionals for the inhomogeneous problem are constructed via two different methods. The first, molecular density functional theory, utilizes the full direct correlation function and an isotropic hard-sphere bridge functional. The second method, a machine learning approach, uses a decomposition of the functional into an isotropic reference part and a mean-field orientational part, where both parts are improved by machine learning techniques. A comparison with the simulation data at hard walls and around hard tracers shows a similar performance of the two functionals. Machine learning strategies are discussed to eliminate residual differences, with the goal of obtaining machine-learning enhanced functionals for the general anisotropic fluid.