A potential distribution induced mapping of free energies for simple fluids

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The potential distribution theorem (PDT) is utilized to construct an effective density, the pseudo-densityρpseudo(z), that enables mapping of the free energies of the uniform fluid exactly onto the nonuniform system values. In addition, a similar quantity, the pseudo-chemical potentialμpseudo(z), is given as the chemical potential produced by the uniform equation of state upon using the nonuniform density ρw(1)(z) as input. The PDT connects three quantities: the work Wins(z) for inserting a test particle into the fluid, the chemical potential μ0 of the bulk fluid, and the nonuniform singlet density ρw(1)(z). We perform Metropolis NVT ensemble Monte Carlo (MC) simulations to obtain the insertion work Wins(z) (via Widom’s particle insertion) and the densities ρw(1)(z). We illustrate the mapping on two simple fluids adsorbed on a hard wall: the Lennard-Jones and the attractive Yukawa fluids. The pseudo-density is determined via an accurate uniform-fluid equation of state for the Lennard-Jones system, and for the Yukawa fluid via direct MC simulations. We characterize the behavior of the effective density and the pseudo-chemical potential vis-à-vis the cases of enhancement and depletion of the fluid density near the wall. These quantities (ρpseudo&μpseudo) are found to exhibit for enhanced adsorption out-of-phase oscillations compared to ρw(1)(z) and βWins(z). For depleted adsorption, we do not observe oscillations and the trends of ρpseudo and μpseudo are in good agreement with those of ρw(1)(z) and βWins. We analyze the differences in behavior in terms of the concavity of the chemical-potential function. We also show the equivalence of the potential distribution theorem to the Euler–Lagrange equation of the density functional theory.