Ordering transitions of weakly anisotropic hard rods in narrow slitlike pores

The effect of strong confinement on the positional and orientational ordering is examined in a system of hard rectangular rods with length $L$ and diameter $D$ ($L>D$) using the Parsons-Lee modification of the second virial density-functional theory. The rods are nonmesogenic $(L/D<3)$ and confined between two parallel hard walls, where the width of the pore ($H$) is chosen in such a way that both planar (particle's long axis parallel to the walls) and homeotropic (particle's long axis perpendicular to the walls) orderings are possible and a maximum of two layers is allowed to form in the pore. In the extreme confinement limit of $H\le 2D$, where only one-layer structures appear, we observe a structural transition from a planar to a homeotropic fluid layer with increasing density, which becomes sharper as $L\to H$. In wider pores $(2D<H<3D)$ planar order with two layers, homeotropic order, and even combined bilayer structures (one layer is homeotropic, while the other is planar) can be stabilized at high densities. Moreover, first-order phase transitions can be seen between different structures. One of them emerges between a monolayer and a bilayer with planar orders at relatively low packing fractions.

Figure 1

Cartoons show the close packing structures of hard rectangular rods with square cross sections in a slitlike pore. Structures of a single homeotropic layer (upper panel), a planar bilayer (middle panel) and a combined bilayer (lower panel) can be realized with the shown $H/D$ and $L/D$ parameters.

Figure 2

Planar-to-homeotropic structural change of hard rods in a narrow slitlike pore $(H/D=2)$, where only one layer of particles is allowed to form. The fraction of hard rods in planar $\left(X_P\right)$ and homeotropic $\left(X_H\right)$ ordering is shown as a function of packing fraction for $L/D=1.9$, 1.99, and 1.999.

Figure 3

Phase diagram of confined hard rods in the packing fraction−shape anisotropy $(\eta -L/D)$ plane for (a) wall-to-wall separations of $H/D=3$ (upper panel); (b) $H/D=2.5$ (lower panel) and $H/D=2.2$ (inset of the lower panel). The curves show the boundaries of the different structures. The following structures are found: uniaxial planar (UP), biaxial planar (BP), homeotropic (H), uniaxial (UT), and biaxial combined two-layer (BT) structures. The symbols designate few stable points of the observed phases. The structures of the hard rods at these symbols are shown in Fig. 4. The biphasic regions are shaded with gray.

Figure 4

Density profiles of the stable structures as a function of $z^{*}=z/D$ in wide $(H/D=3)$ and narrow $(H/D=2.25)$ pores at different packing fractions and shape anisotropies, which can be identified with the help of the shown symbols in Figs. 3 and 5. The local density components ${\rho }_x^{*}\left(z\right)$ (black), ${\rho }_y^{*}\left(z\right)$ (red), and ${\rho }_z^{*}\left(z\right)$ (blue) are shown together in the diagrams. The following stable structures are seen: (a) uniaxial planar $[{\rho }_x^{*}\left(z\right)={\rho }_y^{*}\left(z\right)>{\rho }_z^{*}\left(z\right)]$, (b) biaxial planar $[{\rho }_x^{*}\left(z\right)\ne {\rho }_y^{*}\left(z\right)\ne {\rho }_z^{*}\left(z\right)]$, (c) homeotropic $[{\rho }_x^{*}\left(z\right)={\rho }_y^{*}\left(z\right)\ll {\rho }_z^{*}\left(z\right)]$, (d) uniaxial weakly planar with non-negligible portion of particles in homeotropic order, (e) uniaxial strongly planar with some homeotropically ordered particles at the vicinity of the walls, (f) uniaxial bilayer with homeotropic order at the left wall and planar order at the right wall, (g) bilayer with homeotropic order at the left wall and biaxial planar order at the right wall, (h) uniform monolayer with weak adsorption in planar order at the walls, and (i) uniaxial planar bilayer with few particles in the homeotropic order.

Figure 5

Phase coexistence between monolayer and bilayer fluids in packing fraction−shape anisotropy $(\eta -L/D)$ plane for pore widths $H/D=2.1$, 2.15, 2.2, and 2.25. The density profiles of the uniform and planar phases, which are marked by symbols, can be seen in Figs. 4 and 4.