Primitive subunits: models for cooperative motion in glass-forming liquids. Application to an idealized polymer melt

A family of mechanistic models are proposed to describe cooperative molecular displacements in glass-forming liquids. These models assume that within each cooperative domain motion occurs by sequences of discrete, localized events, but that each of these events involves synchronous displacement of a smaller cluster of neighboring components. The size and properties of this cluster, described as a ‘‘primitive subunit,’’ are assumed to reflect intrinsic details of local structure, not sensitive to external degrees of freedom. However, the length of the sequence of events by which each subunit moves is assumed to be a statistical function of some communal degree of freedom such as the free volume. Two examples are explored for the case of a polymer melt in which idealized conformational rearrangements are constrained by steric interferences. The distribution of lengths for cooperative sequences of events required to remove these interferences is derived as a function of a parameter β related to free volume. It is shown that the mean length of such sequences diverge to infinity for some nonzero (critical) value of β, and that the divergence obeys a scaling law. The divergence is a time-invariant feature of the model, similar in physical significance to the equilibrium phase transition which has been proposed as the underlying basis for the glass transition. However, in the present models it follows from mechanistic constraints, independent of any thermodynamic considerations.