Time-temperature superposition and relaxational behavior in polymeric glasses

It is shown that a free-volume treatment of the relaxational behavior of a polymer can be extended into the glassy region. A modified form of the WLF equation is derived in which the temperature is replaced by T_e, a parameter related to the “frozen” free volume in the glass and defined such that Te = T at equilibrium. Measurements of the isothermal volume contraction of polystyrene and poly(methyl methacrylate) between ?20 and + 95°C are used to estimate the “frozen” free volume and to calculate the temperature dependence of log(a_T) below Tg. The calculated shift factors are compared to experimental values for the glassy state, and good agreement is obtained by selecting an arbitrary, but reasonable, equilibrium glass volume-temperature curve. The slope of this equilibrium glass curve is smaller than the experimental volume curve at some finite cooling rate. The data indicate that the glass is not an “iso-free volume” state and that the relaxation mechanisms in the glass are controlled primarily by the free volume, at least in the vicinity of Tg. A quantitative definition of the role of free volume in the glassy state requires evaluation of the quantities ? log a_T/?V)_T,P and (? log a_T/?T)_v,p; sufficient data of this type are not presently available.