We consider general disordered models of pinning of directed polymers on
a defect line. This class contains in particular the (1+1)-dimensional interface wetting
model, the disordered Poland–Scheraga model of DNA denaturation and other (1 + d)-
dimensional polymers in interaction with flat interfaces. We consider also the case of
copolymers with adsorption at a selective interface. Under quite general conditions,
these models are known to have a (de)localization transition at some critical line in the
phase diagram. In this work we prove in particular that, as soon as disorder is present,
the transition is at least of second order, in the sense that the free energy is differentiable
at the critical line, so that the order parameter vanishes continuously at the transition.
On the other hand, it is known that the corresponding non-disordered models can have
a first order (de)localization transition, with a discontinuous first derivative. Our result
shows therefore that the presence of the disorder has really a smoothing effect on the
transition. The relation with the predictions based on the Harris criterion is discussed.