Routes to the density profile and structural inconsistency

TL;DR

The paper investigates how to obtain accurate one-body density profiles for inhomogeneous classical fluids by leveraging two-body integral equation closures, specifically comparing YBG and LMBW sum rules. By incorporating a closure-based force-DFT framework and optimizing a Verlet-type closure (Modified Verlet) to minimize structural inconsistency between the virial and compressibility routes, the authors demonstrate compressibility-consistent density profiles that agree with Brownian-dynamics simulations in two-dimensional hard-core Yukawa systems. The work shows that the LMBW route is mathematically equivalent to standard DFT yet remains implementable without an explicit excess free-energy functional, enabling a principled optimization of closures to improve predictions in confined geometries and under various external potentials. Overall, the approach provides a route to principal, closure-based density predictions with improved consistency, with potential extensions to dynamics and more complex interparticle interactions. The numerical results highlight the importance of closure tuning for accurately capturing packing effects and confinement-induced structuring in inhomogeneous fluids.

Abstract

Classical density functional theory (DFT) is the primary method for investigations of inhomogeneous fluids in external fields. It requires the excess Helmholtz free energy functional as input to an Euler-Lagrange equation for the one-body density. A variant of this methodology, the force-DFT, uses instead the Yvon-Born-Green equation to generate density profiles. It is known that the latter are consistent with the virial route to the thermodynamics, while DFT is consistent with the compressibility route. In this work we will show an alternative DFT scheme using the Lovett-Mou-Buff-Wertheim (LMBW) equation to obtain density profiles, that are shown to be also consistent with the compressibility route. However, force-DFT and LMBW DFT can both be implemented using a closure relation on the level of the two-body correlation functions. This is proven to be an advantageous feature, opening the possibility of an optimisation scheme in which the structural inconsistency between different routes to the density profile is minimized. (Structural inconsistency is a generalization of the notion of thermodynamic inconsistency, familiar from bulk integral equation studies.) Numerical results are given for the density profiles of two-dimensional systems of hard-core Yukawa particles with a repulsive or an attractive tail, in planar geometry.

Figures (5)

• Figure 1: Selection of external potentials. The softened repulsive hard-wall potential \ref{['external potential contact thm']} is used to test the contact theorem. Its exponential tail is shown in panel A. The soft parts of both confining external potentials \ref{['external potential exp walls']} and \ref{['external potential harmonic trap']} are shown in panel B.
• Figure 2: Testing the contact theorem. We consider two types of HCY particles \ref{['HCY potential']}, both with $\alpha\!=\!2$. Column 1, shows results for $\kappa\!=\!-1.5\!<\!0$, thus an attractive tail. Column 2, shows results for particles with a repulsive tail, with $\kappa\!=\!10\!>\!0$. In panels A, we show reduced pressure curves calculated using both the virial \ref{['virial equation 2D']} and compressibility \ref{['compressibility pressure equation']} equations of state, given by the solid orange and dashed limegreen curves, respectively. The red circles show the results obtained via equation \ref{['specific wall theorem']}, when the input density profile is generated by the YBG equation \ref{['YBG equation']}. The green circles show the same, when the input density profile is generated by the LMBW equation \ref{['LMBW']}. In panels B, we show the contact theorem integrand as solid green lines and the density profile generated by the LMBW equation as dashed green lines, to illustrate how we obtain the circles shown in panels A.
• Figure 3: Optimization of the density profiles using the Modified Verlet closure. In panels A we show density profiles calculated using the YBG and LMBW equations for various values of the optimization parameter, $\alpha_V$. Since the profiles are symmetric about $z\!=\!0$ we show both YBG and LMBW profiles on the same plot for ease of comparison. The left column of panels concern results for $\langle N\rangle\!=\!0.4$, while the right column is for $\langle N\rangle\!=\!0.8$. Panels B show the root-mean-square difference between the profiles obtained using the two different routes as a function of $\alpha_V$. The minimum of both curves is found to be at $\alpha_V\!=\!1.6$. Panels C show the profiles at this optimal value of $\alpha_V$ and demonstrate the improved structural consistency compared with the standard Verlet closure. We also show simulation data as dotted black curves for comparison.
• Figure 4: Bulk pressure optimization using the Modified Verlet closure. In Panel A we show the bulk pressure from the virial \ref{['virial equation 2D']} and compressibility \ref{['compressibility pressure equation']} equations as a function of the bulk density. The dashed seagreen lines show the results obtained using the standard Verlet closure, as in Figure \ref{['fig contact thm']}. Increasing the parameter $\alpha_V$ from 0.8 (the standard Verlet value) to 1.6 leads to a reduction of thermodynamic inconsistency. Virial pressures are shown as solid limegreen lines, while the compressibility pressures are shown in orange. In panel B we show only the pressures for the standard Verlet closure and the Modified Verlet closure with the optimized $\alpha_V\!=\!1.6$. Panels C show zooms of the data from Panel B to focus on different density regimes. We added simulation data as dotted black curves to panels B and C for comparison.
• Figure 5: Application of the optimized closure to a harmonic trap. We show density profiles in the external field \ref{['external potential harmonic trap']}. In panels A the average number of particles per unit length is $\langle N \rangle \!=\! 0.8$, while in panels B its value is $\langle N \rangle \!=\! 1.0$. The first column (in seagreen) shows results using the standard Verlet closure. The second column (in purple) shows results using the Modified Verlet closure for fixed optimized $\alpha_V\!=\!1.6$. The density profiles calculated with the LMBW equation are given by solid green lines and the ones calculated with the YBG equation are in dashed red. The simulation data are given by the dotted black curves. It is clear that in both test-cases the Modified Verlet closure with $\alpha_V\!=\!1.6$ reduces the structural inconsistency in comparison to the results from the standard Verlet closure.