Ordering of rods near planar and curved surfaces

We study the orientational profile of a semi-infinite system of cylinders bounded in two different ways: by a flat and by a curved wall. The latter corresponds to the interior of a spherical shell, where the dimensions of the rods are comparable to the radius of curvature of the container: they have to accomodate to fill the available space, leading to a rich orientation profile. In order to study these problems, we make a mapping onto a three-state Potts model on a semi-infinite lattice, which is solved using a mean-field approach; we fix the boundary conditions on the surface and in the bulk. In the case of a curved surface, the increase in the effective volume interactions towards the bulk, due to compression, is obtained by increasing the nearest neighbor interactions. The mean-field equations are iterated numerically and we obtain various interesting results concerning the free energy and the orientation profile. We show that there is always a first order transition and the stability of the coexisting phases is strongly affected by the surface. When the surface is disordered and the bulk ordered, the profile may present a step that depends on the degree of disorder on the surface, on the rate of increase of the particle interactions and on the surface external potential. The existence of this step may be relevant to applications in nanotechnology.