  We study interfaces in an Allen-Cahn equation, separating two metastable
states. Our focus is on a directional quenching scenario, where a parameter
renders the system bistable in a half plane and monostable in its complement,
with the region of bistability expanding at a fixed speed. We show that the
growth mechanism selects a contact angle between the boundary of the region of
bistability and the interface separating the two metastable states.
Technically, we focus on a perturbative setting in a vicinity of a symmetric
situation with perpendicular contact. The main difficulty stems from the lack
of Fredholm properties for the linearization in translation invariant norms. We
overcome those difficulties establishing Fredholm properties in weighted spaces
and farfield-core decompositions to compensate for negative Fredholm indices.
