Fundamental measure theory for the inhomogeneous hard-sphere system based on Santos' consistent free energy

Based on Santos' general solution for the scaled-particle differential equation [Phys. Rev. E 86, 040102(R) (2012)], we construct a free-energy functional for the hard-sphere system. The functional is obtained by a suitable generalization and extension of the set of scaled-particle variables using the weighted densities from Rosenfeld's fundamental measure theory for the hard-sphere mixture [Phys. Rev. Lett. 63, 980 (1989)]. While our general result applies to the hard-sphere mixture, we specify remaining degrees of freedom by requiring the functional to comply with known properties of the pure hard-sphere system. Both for mixtures and pure systems, the functional can be systematically extended following the lines of our derivation. We test the resulting functionals regarding their behavior upon dimensional reduction of the fluid as well as their ability to accurately describe the hard-sphere crystal and the liquid-solid transition.

Figure 1

Results for the direct correlation function $c\left(r\right)$ of the pure hard-sphere fluid at different packing fractions. In (a) the result from this work, Eq. (20), is compared to data from simulations by Groot et al. [28] for different packing fractions $\eta$. The relative deviations from the result $c_{\text{WBII}}$ obtained with the WBII functional [24] are plotted for (b) $\eta =0.31$ and (c) $\eta =0.47$. The divergence at $r\to 0$ of the result by Santos [25] and Lutsko [26] is in contradiction with the simulations. The result from this work, Eq. (20), is always within $1\%$ of the WBII result.

Figure 2

Pressure of the homogeneous hard-disk fluid in 2D as obtained via dimensional reduction of various functionals (see text for details) compared to the accurate pressure $p_{\text{MCS}}$ due to Solana and coworkers [29].

Figure 3

Vacancy concentration $n_{\text{vac}}$ of the hard-sphere crystal as a function of the packing fraction $\eta$. The results for the functionals ${\Phi }_{2,\lambda =0}^{\text{PY/CS}}$ and ${\Phi }_{2,\lambda =\frac{1}{2}}^{\text{PY/CS}}$ were obtained using unconstrained minimization of the functional while results for the tensorial WBII functional were obtained using both unconstrained and Gaussian minimization [27]. For comparison, we show simulation results from Kwak et al. [32] as well as from Bennett and Alder [33].

Figure 4

Sections through the density distribution around fcc lattice sites along the lattice directions [100], [110], and [111] for the bulk density $\rho {\sigma }^3=1.04$. The $x$ axis denotes the squared distance from the lattice site and the $y$ axis corresponds to the logarithmic density such that a Gaussian distribution would be a straight line. Full symbols with error bars are Monte Carlo simulation data from Ref. [27] and the lines are results from ${\Phi }_{2,\lambda =0}^{\text{PY}}$. The cartoon indicates the lattice directions in the standard fcc unit cell.