Hard disks confined within a narrow channel

TL;DR

This work develops and tests an inhomogeneous Ornstein–Zernike framework with the Percus–Yevick closure and LMBW sum-rule to study hard disks confined in a channel between parallel walls. It demonstrates that dimensional crossover is handled naturally, yielding exact 1D limits and accurately predicting the quasi-1D zigzag ordering as packing increases, with benchmarks provided by an exact quasi-1D mapping to 1D hard rods. The approach offers a principled, DFT-like route to obtain both the one-body density and inhomogeneous two-body correlations under confinement, and it identifies why the PY closure performs exceptionally well in quasi-1D by suppressing tail contributions to the direct correlation in narrow channels. The results establish the method as a powerful tool for confined fluids and suggest pathways to develop new closures that preserve dimensional crossover while remaining accurate in bulk, aided by exact quasi-1D data as a guide.

Abstract

We employ inhomogeneous integral equation theory to investigate the equilibrium properties of hard disks confined to a channel of width $L$ by hard parallel walls. If the channel width is narrowed below two disk diameters, then the system enters a quasi one-dimensional regime for which the particles cannot move past each other. In the limit when $L$ is equal to one particle diameter the system reduces to the one-dimensional bulk along the center of the channel. We study first the dimensional crossover properties of the inhomogeneous Percus-Yevick (PY) integral equation as $L$ is reduced and then investigate the behaviour of a quasi one-dimensional system as the packing of the particles is increased for a fixed value of $L$. We find that the inhomogeneous PY equation is highly accurate for situations of quasi one-dimensional confinement and that it predicts the onset of a structural transition to a zigzag state at higher packing. The excellent performance of this integral equation method and the ease with which it handles confinement-induced dimensional crossover is a consequence of the improved resolution which comes from treating explicitly the inhomogeneous two-body correlation functions.

Figures (8)

• Figure 1: Dimensional crossover imposed by the sequential application of confining potentials. For example, trapping a 3D system of hard spheres between parallel hard walls with a separation $L\!=\!d$ recovers a 2D bulk system of hard disks. Introducing a further pair of confining walls then reduces the system to 1D hard rods on a line. Finally, the 0D limit is obtained when the external boundaries create a cavity which can admit at most one particle.
• Figure 2: Sketch of dimensional crossover from 2D to 1D. The first panel shows a system of hard-disk particles in a two-dimensional slit, with two hard walls separated by a distance $L$ much larger than one particle diameter, $d$. The second panel shows a slit with a wall separation between one and two particle diameters. In this case the channel is too narrow to allow particles to move past each other. We refer to this as the quasi-1D regime. The third panel shows a slit with a wall separation very close to one particle diameter, for which the system reduces to the one-dimensional bulk system of hard rods on a line.
• Figure 3: Dimensional crossover from hard disks to hard rods. Panel A shows, in shades of green, the one-body density profiles for a system of hard-disk particles between two hard walls, as the slit width $L$ is reduced from $5$ to $1.2$ (for particle diameter $d$ set to unity). For all profiles the average number of particles per unit channel length is kept constant at the value $\langle N\rangle\!=\!0.7$. Panel B shows, in the same shades of green, the evolution of the pair distribution function along a line in the center of the channel, namely $g(z_1\!=\!0,z_2\!=\!0,x)$, as the slit width $L$ is reduced toward unity. The bulk two-dimensional hard-disk radial distribution function, $g_{\text{b}}(r_{12})$, evaluated at $\rho_{\text{b}}\!=\!\langle N\rangle/(L-1)$ is shown as a solid light brown line and the exact radial distribution function of hard rods along the $x$-axis, $g_{\text{1D}}(x)$, is shown as a solid pink line.
• Figure 4: Densely packed hard disks in a quasi-1D slit. For a wall separation $L\!=\!1.5$ we show the maximally packed zigzag state, for which the number of particles per unit channel length, $\langle N\rangle$, is equal to $2/\sqrt{3}$. Red and yellow arrows indicate the paths along which we will show results for the inhomogeneous pair distribution function in Fig. \ref{['fig results quasi 1 dim g']}. Path 1 follows the center of the channel, $z\!=\!0$, whereas path 2 is located at $z\!=\!0.25$, corresponding to disks in contact with the upper wall.
• Figure 5: Density profiles in the quasi one-dimensional regime. We show one-body densities for hard disks trapped in a channel with fixed separation length $L\!=\!1.5$ and for increasing packing. Starting from an average number of particles per unit length $\langle N \rangle \!=\! 0.7$ we recover the exact same dashed green curve than already shown previously in panel A of Fig. \ref{['fig results dim reduction']}. The density profiles for $\langle N \rangle \!=\! 0.75, 0.8$ and $0.85$ are shown as solid lines in different shades of green. The density profile for $\langle N \rangle \!=\! 0.9$, shown as a dotted-dashed light green curve, exhibits the most pronounced contact peaks. However, its value at the center of the channel, for $z\!=\!0$, is lower than that for $\langle N \rangle \!=\! 0.85$.