Liquid-crystal patterns of rectangular particles in a square nanocavity

Using density-functional theory in the restricted-orientation approximation, we analyse the liquid-crystal patterns and phase behaviour of a fluid of hard rectangular particles confined in a two-dimensional square nanocavity of side length $H$ composed of hard inner walls. Patterning in the cavity is governed by surface-induced order, capillary and frustration effects, and depends on the relative values of particle aspect ratio $\kappa\equiv L/\sigma$, with $L$ the length and $\sigma$ the width of the rectangles ($L\ge\sigma$), and cavity size $H$. Ordering may be very different from bulk ($H\to\infty$) behaviour when $H$ is a few times the particle length $L$ (nanocavity). Bulk and confinement properties are obtained for the cases $\kappa=1$, 3 and 6. In the confined fluid surface-induced frustration leads to four-fold symmetry breaking in all phases (which become two-fold symmetric). Since no director distorsion can arise in our model by construction, frustration in the director orientation is relaxed by the creation of domain walls (where the director changes by $90^{\circ}$); this configuration is necessary to stabilise periodic phases. For $\kappa=1$ the crystal becomes stable with commensuration transitions taking place as $H$ is varied. In the case $\kappa=3$ the commensuration transitions involve columnar phases with different number of columns. Finally, in the case $\kappa=6$, the high-density region of the phase diagram is dominated by commensuration transitions between smectic structures; at lower densities there is a symmetry-breaking isotropic $\to$ nematic transition exhibiting non-monotonic behaviour with cavity size.