Orientational ordering and layering of hard plates in narrow slitlike pores

We examine the ordering behavior of hard platelike particles in a very narrow, slitlike pore using the Parsons-Lee density functional theory and the restricted orientation approximation. We observe that the plates are orientationally ordered and align perpendicularly (face-on) to the walls at low densities, a first-order layering transition occurs between uniaxial nematic structures having $n$ and $n+1$ layers at intermediate densities, and even a phase transition between a monolayer with parallel (edge-on) orientational order and $n$ layers with a perpendicular one can be detected at high densities. In addition to this, the edge-on monolayer is usually biaxial nematic, and a uniaxial-biaxial nematic phase transition can be also seen at very high densities.

Figure 1

The three possible orientations of the hard plates between two parallel hard walls. The plate is oriented along the $x$, $y$, and $z$ axes from left to right. The $x$ and $y$ orientations are edge-on, while the $z$ orientation is face-on to the walls. $H$ is the pore width, and $L$ and $D$ are the side lengths of the plate.

Figure 2

Two-dimensional representation of the possible structures of hard plates: a face-on ordering of the plates with the walls in three layers (upper panel), an edge-on ordering of the plates with the walls in one layer (middle panel), and a face-on ordering of the plates with the walls in four layers (lower panel). Here we use the following values: $L/D=0.3$ and $H/D=1.3$.

Figure 3

Density profiles of the plates for the three possible orientations for $L/D=0.6$ and $H/D=1.1$: (a) face-on order at $\eta =0.1$ as the majority of the particles are face-on to the walls, (b) uniaxial edge-on order at $\eta =0.6$ as ${\rho }_x={\rho }_y$ and $X_z<1/2$, and (c) biaxial edge-on order at $\eta =0.8$ as ${\rho }_x\ne {\rho }_y$ and $X_z<1/2$. The quantities are dimensionless, i.e., $z^{*}=z/D{\rho }_i^{*}={\rho }_i{D}^3$.

Figure 4

Phase diagram of confined hard plates with $L/D=0.6$. (a) Mole fractions of the plates in the three possible orientations as a function of packing fraction. (b) Borders of the observed phases in packing fraction-pore width plane. The following phases are observed: one-layer face-on (1LFO), one-layer edge-on uniaxial (1LEO), and one-layer edge-on biaxial order (1LBO). The vertical dashed lines delimit the structures from each other in (a), while the lower and upper dashed curves represent the maximal packing fraction of face-on and edge-on monolayers in (b), respectively.

Figure 5

Density profiles of hard plates with $L/D=0.3$: (a, b) two- and three-layer structures in face-on order at $\eta =0.42$ and $H/D=1.04$, (c, d) three-layer structure in face-on and one-layer biaxial in edge-on order at $\eta =0.51$ and $H/D=1.3$, and (e, f) one-layer edge-on biaxial and four-layer structure in face-on order at $\eta =0.62$ and $H/D=1.3$. The quantities are dimensionless, i.e., $z^{*}=z/D$ ${\rho }_i^{*}={\rho }_i{D}^3$.

Figure 6

Phase diagram of confined hard plates with $L/D=0.3$ in packing fraction-pore width plane. The following structures are observed: face-on bilayer (2L), three-layer (3L) face-on, four-layer (4L) face-on, and biaxial edge-on monolayer (1LBO). The biphasic regions are shaded. The dashed curve shows the maximal packing fraction of 2L structure.

Figure 7

Density profiles of hard plates with $L/D=0.1$: (a) five-layer face-on structure at $H/D=0.775$ and $\eta =0.3$, (b) six-layer face-on structure at $H/D=0.775$ and $\eta =0.34$, (c) five-layer face-on structure at $H/D=0.73$ and $\eta =0.455$, and (d) six-layer face-on structure at $H/D=0.73$ and $\eta =0.455$. The quantities are dimensionless, i.e., $z^{*}=z/D$ and ${\rho }_z^{*}={\rho }_z{D}^3$. At these molecular parameters ${\rho }_x^{*}\left(z^{*}\right)={\rho }_y^{*}\left(z^{*}\right)=0$ and ${\rho }_z^{*}\left(z\right)={\rho }_{}^{*}\left(z^{*}\right)$.

Figure 8

Phase diagram of confined hard plates with $L/D=0.2$ (a), 0.15 (b), and 0.1 (c) in packing fraction-pore width plane. Layered structures in face-on order with the walls are observed, where the number of layers is between 2 and 9 ($2L,...,9L$). The layering transition occurs between structures having $n$ and $n+1$ layers. The biphasic regions of the layering transitions are shaded. The dashed curves show the maximal packing fraction of 2L, 3L, and 4L structures from left to right. The dotted curve represents the continuous structural change from $n$-to-$n+1$ layers, where $n=2,3,...7$.