Annealing experiments have been carried out at a few degrees below the melting point of different polyethylenes (LDPE, LLDPE, HDPE), of polypropylene (PP) and of Nylon-6. The heat capacities decrease during the annealing, within a 2-4 min time scale, to a lower value which corresponds to the extrapolated heat capacity values obtained for the cooling cycle when the polymer is cooled from the melt. Heat capacities in the heating cycle following the cooling cycle of PP, Nylon-6 and HDPE have the same value as during the cooling section. This is not the case for LDPE and LLDPE. Exothermic total heat flow in the cooling section following the annealing indicates that the crystallisation takes place during the cooling rather than during the annealing period. The total melting enthalpy measured before and after the annealing cycle is the same. The reversing heat flow shows an excellent fit to the change of the crystallinity measured by small angle scattering of synchrotron radiation during a heating cycle at temperatures below the melting peak. A coupled thermodynamic interaction of the crystalline and the amorphous phases is concluded from this study. This kind of interaction is possible at the lateral end of polymeric chains incorporated into the crystalline phase. This is an indication of the portion of tie molecules in the system, i.e. the portion of fringed micelle type of crystalline morphology with respect to that of folded chain lamellae.