Phase behaviour and correlations of parallel hard squares: from highly confined to bulk systems

Evidence of T-type structures of hard square boards in capillary confinement

We employ Onsager's second virial density functional theory combined with the Parsons-Lee theory within the restricted orientation (Zwanzig) approximation to examine the phase structure of hard square boards of dimensions (L×D×D) uniaxially confined in narrow slabs. Depending on the wall-to-wall separation (H), we predict a number of distinctly different capillary nematic phases, including a monolayer uniaxial or biaxial planar nematic, homeotropic with a variable number of layers, and a T-type structure. We determine that the favored phase is homotropic, and we observe first-order transitions from the homeotropic structure with n layers to n+1 layers as well as from homeotropic surface anchoring to a monolayer planar or T-type structure involving both planar and homeotropic anchoring at the pore surface. By increasing the packing fraction, we further demonstrate a reentrant homeotropic-planar-homeotropic phase sequence in a particular range (i.e., H/D=1.1 and 0.25≤L/D<0.26). We find that the T-type structure is more stable when the pore is wide enough with respect to the planar phase. The enhanced stability of the mixed-anchoring T-structure is unique for square boards and becomes manifest at pore width exceeding L+D. More specifically, the biaxial T-type structure emerges directly from the homeotropic state without intervention of a planar layer structure as observed for other convex particle shapes.

Capillary-driven biaxial planar and homeotropic nematization of hard cylinders

We use the Parsons-Lee modification of Onsager's second virial theory within the restricted orientation (Zwanzig) approximation to analyze the phase behavior of hard cylindrical rods confined in narrow pores. Depending on the wall-to-wall separation we predict a number of distinctly different surface-generated nematic phases, including a biaxial planar nematic with variable number of layers, a monolayer homeotropic, and a hybrid T-type structure (a planar layer combined with a homeotropic one). For narrow pores, we find evidence of two types of second-order uniaxial-biaxial transitions depending on the aspect ratio of the particles. More specifically, we observe a continuous crossover from n to n+1 layers, each with a distinct planar anchoring symmetry as well as first-order transitions from planar to homeotropic surface anchoring. Contrary to the previously studied case of parallelepipeds we find that the surface anchoring transition from planar to homeotropic symmetry occurs at much lower overall rod packing fractions. This renders the observation of homeotropic capillary nematics much more realistic in experimental systems of strongly confined anisotropic colloids. Unlike confined parallelepipeds, cylindrical rods gradually increase the number of the nematic planar layers (without any phase transitions). However, a weakly first-order transition was observed between two planar structures with n and n+1 layers in wide pores and longer rods. In addition, the cylindrical rods exhibit a first-order transition from the homeotropic structure to the uniaxial (or biaxial) T phase that has not been observed in confined hard parallelepipeds. We further demonstrate a reentrant uniaxial-biaxial-uniaxial-biaxial phase sequence for confined cylinders at small aspect ratio. Our results also clearly demonstrate that stable T-type surface ordering is a subtle capillary effect that only becomes manifest in sufficiently narrow pores away from the two-dimensional bulk limit.

Highly confined mixtures of parallel hard squares: A density-functional-theory study

Using the fundamental-measure density-functional theory, we study theoretically the phase behavior of extremely confined mixtures of parallel hard squares in slit geometry. The pore width is chosen such that configurations consisting of two consecutive big squares, or three small squares, in the transverse direction, perpendicular to the walls, are forbidden. We analyze two different mixtures with edge lengths of species selected so as to allow or forbid one big plus one small square to fit into the channel. For the first mixture we obtain first-order transitions between symmetric and asymmetric packings of particles: Small and big squares are preferentially adsorbed at different walls. Asymmetric configurations are shown to lead to more efficient packing at finite pressures. We argue that the stability region of the asymmetric phase in the pressure-composition plane is bounded so that the symmetric phase is stable at low and very high pressure. For the second mixture, we observe strong demixing between phases which are rich in different species. Demixing occurs in the lateral direction, i.e., the dividing interface is perpendicular to the walls, and phases exhibit symmetric density profiles. The possible experimental realization of this behavior (which in practical terms is precluded by jamming) in strictly two-dimensional systems is discussed. Finally, the phase behavior of a mixture with periodic boundary conditions is analyzed and the differences and similarities between the latter and the confined system are discussed. We claim that, although exact calculations exclude the existence of true phase transitions in (1+ε)-dimensional systems, density-functional theory is still successful in describing packing properties of large clusters of particles.

Positional ordering of hard adsorbate particles in tubular nanopores

The phase behavior and structural properties of a monolayer of hard particles is examined in such a confinement where the adsorbed particles are constrained to the surface of a narrow hard cylindrical pore. The diameter of the pore is chosen such that only first- and second-neighbor interactions occur between the hard particles. The transfer operator method of [Percus and Zhang, Mol. Phys. 69, 347 (1990)MOPHAM0026-897610.1080/00268979000100241] is reformulated to obtain information about the structure of the monolayer. We have found that a true phase transition is not possible in the examined range of pore diameters. The monolayer of hard spheres undergoes a structural change from fluidlike order to a zigzaglike solid one with increasing surface density. The case of hard cylinders is different in the sense that a layering takes place continuously between a low-density one-row and a high-density two-row monolayer. Our results reveal a clear discrepancy with classical density functional theories, which do not distinguish smecticlike ordering in bulk from that in narrow periodic pores.

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≤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→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.