  We present a detailed analysis of three models predicting the properties of
non-uniform matter in the crust of neutron stars: the compressible liquid-drop
model, the fourth order Extended Thomas Fermi (ETF) method, and ETF plus
Strutinsky integral (ETFSI) correction. The former treats the nuclear clusters
as uniform hard spheres, the second takes into account the density distribution
which can be different for neutrons and protons, and the last one includes the
proton shell effects within the Strutinsky approach. The purpose of this work
is to understand the importance of the improvements in the nuclear modeling and
to analyze the quantities which are the most sensitive to them. We find that
thermodynamic quantities such as pressure, energy and chemical potential, as
well as the electron fraction, are in very good agreement among the three
models. This confirms previous results where we have shown that the improvement
in the finite-size description of the nuclear clusters has a small impact on
these quantities, since they are mainly constrained by the bulk properties. The
refinements in the finite-size modeling are shown to impact mostly the
composition of the nuclear clusters ($Z_{cl}$, $N_{cl}$) in an ordering which
ranks according to the leptodermous expansion. This analysis is performed
considering both the r-cluster and the e-cluster representations. The proton
shell effects are shown to stabilize $Z_{cl}$, which consequently impacts the
neutron number $N_{cl}$ as well.
