  Recent experiments have demonstrated that the glass transition temperature of
thin polymer films can be shifted as compared to the same polymer in the bulk,
the amplitude and the sign of this effect depending on the interaction between
the polymer and the substrate. A model has been proposed recently for
explaining these effects in two limiting cases: suspended films and strongly
adsorbed films. We extend here this model for describing the cross-over between
these two situations. We show here how, by adjusting the strength of the
adsorption, one can control the glass transition temperature of thin polymer
films. In particular, we show that the shift of glass transition temperature,
refered to that of a suspended film, varies like $\epsilon^{1/\gamma_2}$ where
$\epsilon$ is the adsorption energy per monomer and $\gamma_2$ is the critical
exponent for the mass of aggregates in the 2D percolation problem. We show also
that the interaction leads to an increase of $T_g$ for $\epsilon$ as small as a
few $0.01 T$ where $T$ is the thermal energy, and that this increase saturates
at the value corresponding to strongly interacting films for adsorption
energies of order $0.5 T$.
