  The glass transition is a long-standing unsolved problem in materials
science. For polymers, our understanding of glass-formation is particularly
poor due to the added complexity of chain connectivity and flexibility;
structural relaxation of polymers thus involves a complex interplay between
intra- and inter-molecular cooperativity. Here we study how the glass
transition temperature Tg varies with molecular weight M for different polymer
chemistries and chain flexibilities. We find that Tg(M) is controlled by the
average mass (or volume) per conformational degree of freedom, and that a
`local' molecular relaxation (involving a few conformers) controls the
larger-scale cooperative alpha relaxation responsible for Tg. We propose that
dynamic facilitation where a `local' relaxation facilitates adjacent
relaxations, leading to hierarchical dynamics, can explain our observations
including logarithmic Tg(M) dependences. Our study provides a new understanding
of molecular relaxations and the glass transition in polymers, which paves the
way for predictive design of polymers based on monomer-scale metrics.
