Mean-field theory of inhomogeneous fluids

S. M. Tschopp, H. D. Vuijk, A. Sharma, and J. M. Brader

Phys. Rev. E 102, 042140 – Published 29 October 2020

Abstract

The Barker-Henderson perturbation theory is a bedrock of liquid-state physics, providing quantitative predictions for the bulk thermodynamic properties of realistic model systems. However, this successful method has not been exploited for the study of inhomogeneous systems. We develop and implement a first-principles “Barker-Henderson density functional,” thus providing a robust and quantitatively accurate theory for classical fluids in external fields. Numerical results are presented for the hard-core Yukawa model in three dimensions. Our predictions for the density around a fixed test particle and between planar walls are in very good agreement with simulation data. The density profiles for the free liquid vapor interface show the expected oscillatory decay into the bulk liquid as the temperature is reduced toward the triple point, but with an amplitude much smaller than that predicted by the standard mean-field density functional.

Received 12 August 2020
Accepted 13 October 2020

DOI:https://doi.org/10.1103/PhysRevE.102.042140

Physics Subject Headings (PhySH)

Classical statistical mechanics Interface & surface thermodynamics

Statistical Physics & Thermodynamics

Authors & Affiliations

S. M. Tschopp ¹, H. D. Vuijk², A. Sharma ², and J. M. Brader¹

¹Department of Physics, University of Fribourg, CH-1700 Fribourg, Switzerland
²Leibniz-Institut für Polymerforschung Dresden, Institut Theorie der Polymere, 01069 Dresden, Deutschland

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 102, Iss. 4 — October 2020

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Phase diagram for $α = 1.8$ . Standard mean-field theory (broken red like), BH theory (full black line), and MC simulation data taken from Ref. [14]. The triangle indicates the state point at which we show the radial distribution in Fig. 2 and the arrow indicates the path taken when calculating the density profiles in Fig. 5.
Reuse & Permissions
Figure 2
Test particle. Comparison of the radial distribution function calculated using the test particle method with simulation data at $κ = 1.111$ and $ρ_{b} = 0.8$ (marked with a triangle in Fig. 1). Green circles: MC simulation. Full black line: BH functional. Broken red line: SMF theory. Dashed blue line: The density of pure hard spheres ( $κ = 0, ρ_{b} = 0.8$ ) calculated using the Rosenfeld functional. Insets (a) and (b) focus on the second peak and contact value, respectively.
Reuse & Permissions
Figure 3
Test particle. The companion to Fig. 2 showing the individual contributions to the one-body direct correlation function appearing in Eq. (15). Full black lines: BH functional. Broken red line: SMF theory. The blue lines are a guide for the eye to show how the maxima and minima of these functions match up with the oscillations in the radial distribution function.
Reuse & Permissions
Figure 4
Planar slit. The density between two hard walls located at $z = 0$ and 10 for $κ = 0.5$ . Green circles: GCMC data at $μ = - 1$ . DFT profiles calculated at $μ = - 1$ are given by the broken blue line (BH functional) and the broken red line (SMF functional). DFT profiles calculated by adjusting $μ$ such that the average number of particles in the system matches that of simulation are given by the full black line (BH functional) and the full red line (SMF functional).
Reuse & Permissions
Figure 5
Planar slit. The density between two hard walls located at $z = 0$ and 10 for $κ = 0.75$ . Green circles: GCMC data at chemical potentials $μ = - 2.50, - 2.25, - 2.00, - 1.50$ , and $- 1.00$ (moving along the arrow marked in Fig. 1). Broken blue line: BH functional at the same chemical potentials as used in the simulation. Full black line: BH functional profiles calculated by adjusting $μ$ such that the average number of particles in the system matches that of simulation.
Reuse & Permissions
Figure 6
Free interface. Density profiles at the liquid-vapor interface from (a) the SMF functional and (b) the BH functional. As the two theories have different binodals (see Fig. 1) we compare profiles with equal values of the coexisting liquid density. (c) Focuses on the decay of the density into the bulk liquid for the state point closest to the triple point. The arrows are intended to help the reader better see which of the oscillations in (a) and (b) are being shown in (c).
Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics