Fundamental-measure free-energy density functional for hard spheres: Dimensional crossover and freezing

Abstract

A geometrically based fundamental-measure free-energy density functional unified the scaled-particle and Percus-Yevick theories for the hard-sphere fluid mixture. It has been successfully applied to the description of simple ͑''atomic''͒ three-dimensional ͑3D͒ fluids in the bulk and in slitlike pores, and has been extended to molecular fluids. However, this functional was unsuitable for fluids in narrow cylindrical pores, and was inadequate for describing the solid. In this work we analyze the reason for these deficiencies, and show that, in fact, the fundamental-measure geometrically based theory provides a free-energy functional for 3D hard spheres with the correct properties of dimensional crossover and freezing. After a simple modification of the functional, as we propose, it retains all the favorable Dϭ3 properties of the original functional, yet gives reliable results even for situations of extreme confinements that reduce the effective dimensionality D drastically. The modified functional is accurate for hard spheres between narrow plates (Dϭ2), and inside narrow cylindrical pores (Dϭ1), and it gives the exact excess free energy in the Dϭ0 limit ͑a cavity that cannot hold more than one particle͒. It predicts the ͑vanishingly small͒ vacancy concentration of the solid, provides the fcc hard-sphere solid equation of state from closest packing to melting, and predicts the hard-sphere fluid-solid transition, all in excellent agreement with the simulations.

Free Energy Calculations of Crystalline Hard Sphere Complexes Using Density Functional Theory

“…15,16 Figure 2 shows the excess free energy as a function of the packing fraction for AB, AB 2 , and AB 13 structures. Simulation results (dashed line) are extracted from Jackson et al 18 For AB 13 structure the calculated excess free energies are very close to simulation results at lower packing fractions. The percentage discrepancy is less that 1% at η ≈ 0.54.…”

“…The pressure shows an excellent agreement with the simulation data of the AB and AB 2 structures. For AB 13 , there is a noticeable deviation from the simulation at higher packing fractions. The corresponding simulation data (dashed lines) are also taken from ref 18.…”

“…15 3. COMPARISON OF FMT RESULTS OF AB, AB 2 , and AB 13 WITH SIMULATIONS The Helmholtz free energy for a given structure is obtained by minimizing the free energy functional, eq 4, with respect to the normalized Gaussian parameters as mentioned previously. This is a two-dimensional minimization for binary crystals.…”

“…In section 2, we summarize the basic ingredients of the "White Bear" functional of the fundamental measure theory used in our work. Then in section 3 the results of AB 2 , AB 13 , and AB stable hard sphere crystals are compared with simulations. In section 4 the hard sphere models of the above-mentioned crystals in the Cu−Zr system are presented to pave the way for another paper where the calculation of Cu−Zr phase diagram is presented.…”

Crystal Nucleation in the Hard-Sphere System Revisited: A Critical Test of Theoretical Approaches

“…40 This approach has the advantage that it predicts the interfacial properties of the liquid-fcc interface, whereas the far simpler continuum models used in the present paper need the interfacial properties as an input. Unfortunately, while the interfacial free energies, 0.69kT/σ 2 and 0.66kT/σ 2 , the two variants of the DFT-FM 40 predicted for the equilibrium crystal-fluid interface, are fairly accurate for approaches based on first principles, these values significantly exceed the best value (0.56 ( 0.02)kT/σ 2 from atomistic simulations. Accordingly, a direct comparison of the DFT-MF results with our predictions for the nucleation barrier (obtained with accurate interfacial data) or with the respective MC results might prove inconclusive.…”