Layering, freezing, and re-entrant melting of hard spheres in soft confinement

Confinement can have a dramatic effect on the behavior of all sorts of particulate systems, and it therefore is an important phenomenon in many different areas of physics and technology. Here, we investigate the role played by the softness of the confining potential. Using grand canonical Monte Carlo simulations, we determine the phase diagram of three-dimensional hard spheres that in one dimension are constrained to a plane by a harmonic potential. The phase behavior depends strongly on the density and on the stiffness of the harmonic confinement. While we find the familiar sequence of confined hexagonal and square-symmetric packings, we do not observe any of the usual intervening ordered phases. Instead, the system phase separates under strong confinement, or forms a layered re-entrant liquid phase under weaker confinement. It is plausible that this behavior is due to the larger positional freedom in a soft confining potential and to the contribution that the confinement energy makes to the total free energy. The fact that specific structures can be induced or suppressed by simply changing the confinement conditions (e.g., in a dielectrophoretic trap) is important for applications that involve self-assembled structures of colloidal particles.

Figure 1

Phase diagram in the spring constant-density representation. Every point represents the result of a single simulation run. Black crosses ($\times$): liquid. Red triangles ($▵$): hexagonally packed. Blue squares ($\square$): square packed (structures shown in Fig. 2). Approximate phase boundaries are indicated with solid lines. Dashed lines indicate the stable phases in the limit $k\to \infty$ (from Fig. 3).

Figure 2

Simulation snapshots at different densities (for $k=100$) and “ideal” schematic representations (lower right) of the spatial arrangement of the particles in the square ($\square$) and hexagonally ($▵$) packed structures.

Figure 3

Normalized energies per unit area of the hexagonal (red lines, $▵$) and square (blue lines, $\square$) packed structures in the limit $k\to \infty$.

Figure 4

Density-dependent re-entrant melting between the 1$▵$ and 2$\square$ phases, for $k=100$ and different sizes of the simulation box ($L=L_x=L_y$). Red stars: $L=50$. Green diamonds: $L=20$. Blue circles: $L=10$. The insets give the probability distribution of observing a given density at coexistence chemical potential on the same density scale as the main plot.

Figure 5

Particle distribution along the axis of confinement ($z$) for $k=100$ and different densities. Red triangles: 1$▵$, $\rho =1.05$. Green crosses: liquid, $\rho =1.23$. Yellow diamonds: liquid, $\rho =1.43$. Blue squares: 2$\square$, $\rho =1.58$.