On the computation of fundamental measure theory in pores with cylindrical symmetry

Abstract

Classical density functional theories usually separate the formulation of the excess Helmholtz free energy in hard-body and energetic contributions. Fundamental measure theories FMTs have emerged as the preferred choice to account for the former contribution. The evaluation of geometrically weighted densities convolutions arisen in FMT for hard spheres in long cylindrical cavities is addressed in this paper. Previously, Malijevský J. Chem. Phys. 126, 134710 2007 reported expressions containing elliptic integrals for the kernels of the convolutions involving scalar and vectorial weights. Here, the set of kernels is extended to second and third order tensorial weights that introduce desirable dimensional crossover properties to the evaluation of the excess free energy. An alternative formulation for the convolutions, which greatly facilitates their computation, is also proposed. Integrals of the original kernels arise in this way and a set of expressions for them, again expressed in terms of elliptic integrals, is presented here. With the aim of providing a computationally simple framework to evaluate equilibrium density profile with cylindrical symmetry, a procedure based on direct minimization of the discretized grand potential energy, rather than employing the Euler–Lagrange equilibrium conditions, is discussed and used to identify differences between two FMT formulations, including or not second order tensorial kernels in very narrow cylindrical pores.