  A theoretical analysis is presented of a nematic liquid crystal confined
between substrates pat- terned with squares that promote vertical and planar
alignment. Two approaches are used to eluci- date the behavior across a wide
range of length scales: Monte Carlo simulation of hard particles and
Frank-Oseen continuum theory. Both approaches predict bistable degenerate
azimuthal alignment in the bulk along the edges of the squares; the continuum
calculation additionally reveals the possi- bility of an anchoring transition
to diagonal alignment if the polar anchoring energy associated with the pattern
is sufficiently weak. Unlike the striped systems previously analyzed, the Monte
Carlo simulations suggest that there is no "bridging" transition for
sufficiently thin cells. The extent to which these geometrically patterned
systems resemble topographically patterned substrates, such as square wells, is
also discussed.
