Orientational ordering of confined hard rods: The effect of shape anisotropy on surface ordering and capillary nematization

We examine the ordering properties of rectangular hard rods with length $L$ and diameter $D$ at a single planar wall and between two parallel hard walls using the second virial density-functional theory. The theory is implemented in the three-state Zwanzig approximation, where only three mutually perpendicular directions are allowed for the orientations of hard rods. The effect of varying shape anisotropy is examined at $L/D=10,15,\mathrm{and}20$. In contact with a single hard wall, the density profiles show planar ordering, damped oscillatory behavior, and a wall-induced surface ordering transition below the coexisting isotropic density of a bulk isotropic-nematic $\left(I\text{−}N\right)$ phase transition. Upon approaching the coexisting isotropic density, the thickness of the nematic film diverges logarithmically, i.e., the nematic wetting is complete for any shape anisotropy. In the case of confinement between two parallel hard walls, it is found that the continuous surface ordering transition depends strongly on the distance between confining walls $H$ for $H<L$, while it depends weakly on $H$ for $H>L$. The minimal density at which a surface ordering transition can be realized is located at around $H\sim 2D$ for all studied shape anisotropies due to the strong interference effect between the two hard walls. The first-order $I\text{−}N$ phase transition of the bulk system becomes a surface ordered isotropic $I_B$ to capillary nematic $N_B$ phase transition in the slit pore. This first-order $I_B\text{−}{N}_B$ transition weakens with decreasing pore width and terminates in a critical point for all studied shape anisotropies.

Figure 1

Schematic representation of the rectangular hard rods confined in a slit pore. Particles are allowed to orient along the $x$, $y$, and $z$ axes (Zwanzig approximation). Here $H$ is the wall-to-wall separation and $L$ and $D$ are the length and the diameter of hard rods, respectively.

Figure 2

Isotropic-nematic phase boundary of hard rods in bulk. Coexisting isotropic $I$ and nematic $N$ densities are shown as a function of aspect ratio $L/D$. The inset depicts the nematic order parameter $S_x$ of the coexisting nematic phase. Here ${\rho }^{*}$ is the reduced number density $({\rho }^{*}=\frac{N}{V}{D}^3)$.

Figure 3

Density profiles and order parameters of the hard rod fluid in contact with a single hard wall that is located at $z^{*}=0$. The aspect ratio $L/D$ is chosen to be 10. (a) and (b) Obtained at $\beta \mu =-2.8$, where the phase is isotropic and the bulk component densities (dotted line) are equal to $4.584\times {10}^{-3}$. (c) and (d) The case $\beta \mu =-2.7$. This phase is biaxial isotropic and the corresponding bulk component densities (dotted line) are $4.712\times {10}^{-3}$. (e) and (f) The reduced chemical potential is $-2.2$, where the bulk component densities are $5.368\times {10}^{-3}$.

Figure 4

Fitted surface tensions ${\gamma }_{\mathrm{WI}}$, ${\gamma }_{\mathrm{WN}}$, and ${\gamma }_{\mathrm{IN}}$ of the hard-wall–isotropic, hard-wall–nematic, and $I\text{−}N$ fluid interfaces, respectively. The calculated surface tensions (solid squares) are fitted by the power law $\gamma \left(x\right)=a{x}^b$ (solid lines), where $x=L/D$. The surface tensions are dimensionless quantities, i.e., ${\gamma }_{\mathrm{WI}}^{*}=\beta {\gamma }_{\mathrm{WI}}{D}^2$, ${\gamma }_{\mathrm{WN}}^{*}=\beta {\gamma }_{\mathrm{WN}}{D}^2$, and ${\gamma }_{\mathrm{IN}}^{*}=\beta {\gamma }_{\mathrm{IN}}{D}^2$.

Figure 5

Density profiles and order parameters of confined hard rods $(L/D=10)$ between two hard walls at reduced wall separation $h^{*}=H/D=35$. (a) and (b) Obtained at $\beta \mu =-3.48$, where the phase is isotropic. (c) and (d) The case $\beta \mu =-2.51$, where the phase is biaxial near the walls (${\rho }_x^{*}\ne {\rho }_y^{*}$) but isotropic in the middle of the pore. (e) and (f) The reduced chemical potential is $-2$, where the phase is biaxial nematic.

Figure 6

Phase diagram of confined hard rods in the density–wall-to-wall separation plane. The $I_B\text{−}{N}_B$ (thick lines) and $U\text{−}B$ (thin lines) phase transition curves are shown. The values of the aspect ratios are (a) $L/D=10$, (b) $L/D=15$, and (c) $L/D=20$. The resulting phase diagrams are shown together in different units in (d).

Figure 7

Critical pore width $H_c/L$ as a function of $D/L$. The stars correspond to our numerical results, while the closed circle is the result of van Roij et al. for $D/L\to 0$ [11]. The fitted equation (solid line) is given by $H_c/L=2.08(1+a{x}^b)$, where $x=D/L$, $a=4.24857$, and $b=0.89038$.