We study binary blends of asymmetric diblock copolymers (AB/AC) in selective solvents with a meso-scale model. We investigate the morphological transitions induced by the concentration of the AC block copolymer and the difference in molecular weight between the AB and AC copolymers, when segments B and C exhibit hydrogen-bonding interactions. To the best of our knowledge, this is the first work modeling mixtures of block copolymers with large differences in molecular weight. The co-assembly mechanism localizes the AC molecules at the interface of A and B domains, and induces the swelling of the B-rich domains. The coil size of the large molecular weight block copolymer, depends only on the concentration of the short block copolymer (AC or AB), regardless of the B-C interactions. However, the B-C interactions control the morphological transitions that occur in these blends.