  We use the Dynamic Density-Functional Formalism and the Fundamental Measure
Theory as applied to a fluid of parallel hard squares to study the dynamics of
heterogeneous growth of non-uniform phases with columnar and crystalline
symmetries. The hard squares are (i) confined between soft repulsive walls with
square symmetry, or (ii) exposed to external potentials that mimic the presence
of obstacles with circular, square, rectangular or triangular symmetries. For
the first case the final equilibrium profile of a well commensurated cavity
consists of a crystal phase with highly localized particles in concentric
square layers at the nodes of a slightly deformed square lattice. We
characterize the growth dynamics of the crystal phase by quantifying the
interlayer and intralayer fluxes and the non-monotonicity of the former, the
saturation time, and other dynamical quantities. The interlayer fluxes are much
more monotonic in time, and dominant for poorly commensurated cavities, while
the opposite is true for well commensurated cells: although smaller, the time
evolution of interlayer fluxes are much more complex, presenting strongly
damped oscillations which dramatically increase the saturation time. We also
study how the geometry of the obstacle affects the symmetry of the final
equilibrium non-uniform phase (columnar vs. crystal). For obstacles with
fourfold symmetry, (circular and square) the crystal is more stable, while the
columnar phase is stabilized for obstacles without this symmetry (rectangular
or triangular). We find that, in general, density waves of columnar symmetry
grow from the obstacle. However, additional particle localization along the
wavefronts gives rise to a crystalline structure which is conserved for
circular and square obstacles, but destroyed for the other two obstacles where
columnar symmetry is restored.
