  Ferroelectrics form domain patterns that minimize their energy subject to
imposed boundary conditions. In a linear, constrained theory, that neglects
domain wall energy, periodic domain patterns in the form of multi-rank
laminates can be identified as minimum-energy states. However, when these
laminates (formed in a macroscopic crystal) comprise domains that are a few
nanometers in size, the domain-wall energy becomes significant, and the
behaviour of laminate patterns at this scale is not known. Here, a phase-field
model, which accounts for gradient energy and strain energy contributions, is
employed to explore the stability and evolution of the nanoscale multi-rank
laminates. The stress, electric field, and domain wall energies in the
laminates are computed. The effect of scaling is also discussed. In the absence
of external loading, stripe domain patterns are found to be lower energy states
than the more complex, multi-rank laminates, which mostly collapse into simpler
patterns. However, complex laminates can be stabilized by imposing external
loads such as electric field, average strain and polarization. The study
provides insight into the domain patterns that may form on a macroscopic single
crystal but comprising of nanoscale periodic patterns, and on the effect of
external loads on these patterns.
