Comparative study of force-based classical density functional theory

We reexamine results obtained with the recently proposed density functional theory framework based on forces (force-DFT) [S. M. Tschopp et al., Phys. Rev. E 106, 014115 (2022)]. We compare inhomogeneous density profiles for hard sphere fluids to results from both standard density functional theory and from computer simulations. Test situations include the equilibrium hard sphere fluid adsorbed against a planar hard wall and the dynamical relaxation of hard spheres in a switched harmonic potential. The comparison to grand canonical Monte Carlo simulation profiles shows that equilibrium force-DFT alone does not improve upon results obtained with the standard Rosenfeld functional. Similar behavior holds for the relaxation dynamics, where we use our event-driven Brownian dynamics data as benchmark. Based on an appropriate linear combination of standard and force-DFT results, we investigate a simple hybrid scheme which rectifies these deficiencies in both the equilibrium and the dynamical case. We explicitly demonstrate that although the hybrid method is based on the original Rosenfeld fundamental measure functional, its performance is comparable to that of the more advanced White Bear theory.

Figure 1

The equation of state of the hard sphere fluid is shown as obtained from the Percus-Yevick approximation both via the compressibility and the virial route as well as from GCMC simulations (as indicated). Thereby, ${\rho }_b$ denotes the bulk density and $\rho \left(0^{+}\right)=\beta P$ is the contact density at the hard wall, which can be associated with the bulk pressure $P$. The upper scales illustrate differences in the chemical potential with respect to the simulation values ${\mu }_{\mathrm{sim}}$ that result from the approximative equations of state via the compressibility (${\mu }_c$) and virial (${\mu }_v$) route (analytical expressions are given in Appendix). Therefore, to yield a valid comparison of the density profiles, $\mu$ has to be tuned appropriately in the GCMC simulation to match the considered bulk densities of the standard and force-DFT results, which is illustrated by the gray vertical lines.

Figure 2

Density profiles $\rho \left(z\right)$ of a hard sphere fluid at a planar hard wall are shown for values ${\rho }_b=0.4890{\sigma }^{-3}$ (a), ${\rho }_b=0.6032{\sigma }^{-3}$ (b), and ${\rho }_b=0.6908{\sigma }^{-3}$ (c) of the bulk density. We compare the results of standard (orange) and force-DFT (blue) to numerically exact density profiles from GCMC simulations (gray). For each value of $\mu$, the absolute error $\mathrm{\Delta }\rho \left(z\right)$ of the density profiles compared to the simulation result is shown in the respective bottom panel, and the inset plot zooms in on the differences of the two DFT routes close to the hard wall. The simulations were set up to yield the same bulk density as in the DFT results via an appropriate choice of the chemical potential (cf. Fig. 1 and Table 1) for a systematic comparison of the resulting contact densities.

Figure 3

Time evolution of the density profile $\rho \left(z\right)$ of a hard sphere fluid in a harmonic external potential $V_{\mathrm{ext}}\left(z\right)=A{(z-5\sigma )}^2$ after switching its strength from $A=0.75k_{B}T/{\sigma }^2$ to $A=0.5k_{B}T/{\sigma }^2$ at time $t=0$. The relaxation dynamics calculated with standard (orange) and force-DDFT (blue) are shown for $t/\tau =0,0.05,0.1,0.2,0.5,1$ and are compared to EDBD simulation results (gray). The initial and final equilibrium profiles (silver) as obtained via MC simulations for both values of $A$ are indicated in each panel for reference.

Figure 4

Hybrid DFT density profiles $\rho \left(z\right)$ (purple) for a hard sphere fluid at a hard wall are compared to simulation results as in Fig. 2 (the standard Rosenfeld DFT is replotted in orange). In most parts of the system, this combination of standard and force-DFT via Eq. (10) enables a systematic improvement of the resulting density profile while retaining the Rosenfeld FMT treatment of $F_{\mathrm{exc}}\left[\rho \right]$. The largest discrepancy to the numerical GCMC density profiles (gray) still occurs in the vicinity of the first density maximum. For comparison, standard DFT results for the superior White Bear (olive) and White Bear MkII (cyan) functionals are depicted, and an error comparable to hybrid Rosenfeld DFT is found.

Figure 5

Hybrid DDFT density profiles $\rho \left(z\right)$ (purple) via Eq. (10) for the relaxation of a hard sphere fluid in a harmonic potential as in Fig. 3. The time evolution is again compared to EDBD simulation results (gray) and the initial and final equilibrium profiles are indicated for reference (silver). As in Fig. 4, the combination procedure (10) of standard and force-DDFT yields much better results than the individual routes alone.