  In this paper, we investigate a model for a $1+1$ dimensional
self-interacting and partially directed self-avoiding walk, usually referred to
by the acronym IPDSAW. The interaction intensity and the free energy of the
system are denoted by $\beta$ and $f$, respectively. The IPDSAW is known to
undergo a collapse transition at $\beta_c$. We provide the precise asymptotic
of the free energy close to criticality, that is we show that
$f(\beta_c-\epsilon)\sim \gamma \epsilon^{3/2}$ where $\gamma$ is computed
explicitly and interpreted in terms of an associated continuous model. We also
establish some path properties of the random walk inside the collapsed phase
$(\beta>\beta_c)$. We prove that the geometric conformation adopted by the
polymer is made of a succession of long vertical stretches that attract each
other to form a unique macroscopic bead, we identify the horizontal extension
of the random walk inside the collapsed phase and we establish the convergence
of the rescaled envelope of the macroscopic bead towards a deterministic Wulff
shape.
