  Polar nematic liquid crystals are new classes of condensed-matter states
where the inversion symmetry common to the traditional apolar nematics is
broken. Establishing theoretical descriptions for the novel phase states is an
urgent task. Here, we develop a Landau-type mean-field theory for both the
achiral and chiral ferroelectric nematics. In the polar nematic states, the
inversion symmetry breaking adds two new contributions: an additional odd
elastic term (corresponding to the flexoelectricity in symmetry) to the
standard Oseen-Frank free energy and an additional Landau term relating to the
gradient of local polarisation. As a general necessity, the coupling between
the scalar order parameter and polarisation order is further considered. In the
chiral and polar nematic state, we reveal that the competition between the
twist elasticity and polarity dictates effective compressive energy arising
from the quasi-layer structure. The polarisation gradient is an essential term
for describing the ferroelectric nature of the systems. The approaches provide
theoretical foundations for testing and predicting polar structures in emerging
polar liquid crystals.
