Scaling behavior of free-volume holes in polymers probed by positron annihilation

Positron lifetimes in polybutadiene (PB) and polypropylene (PP) are measured as functions of temperature in the range 95–305 K and 95–370 K, respectively. From the variations of ortho-positronium (o-Ps) lifetime ${\mathrm{\tau }}_3$, we have determined the glass transition temperatures ${\mathit{T}}_{\mathit{g}}$ in PB and PP and the excluding temperature ${\mathit{T}}_{\mathit{e}}$ in PB, beyond which the o-Ps lifetime ${\mathrm{\tau }}_3$ in PB is saturated. From the analogy of the temperature-dependent o-Ps intensity ${\mathit{I}}_3$ in PB and PP, high- and low-density polyethylene (HDPE and LDPE), we propose that above the glass transition temperature ${\mathit{T}}_{\mathit{g}}$ the o-Ps intensity ${\mathit{I}}_3$ exhibits the scaling relation ${\mathit{I}}_3$∼(T-${\mathit{T}}_{\mathit{g}}$ ${)}^{\mathrm{\beta }}$ in these polymers. The data also indicate the existence of a crossover temperature ${\mathit{T}}_{\mathit{c}}$, located above ${\mathit{T}}_{\mathit{g}}$, where the β value changes greatly. When ${\mathit{T}}_{\mathit{g}}$<T≤${\mathit{T}}_{\mathit{c}}$, β has the order of magnitude of ${10}^{\mathrm{-}2}$. When ${\mathit{T}}_{\mathit{c}}$≤T<${\mathit{T}}_{\mathit{e}}$, β=0.346±0.035 and 0.297±0.037 for linear polymers PB and PP. Similar features are also observed for previous results of HDPE and LDPE [D. Lin and S. J. Wang, J. Phys. Condens. Matter 4, 3331 (1992)]. The β values in the region of ${\mathit{T}}_{\mathit{c}}$≤T<${\mathit{T}}_{\mathit{e}}$ are found to be 0.336±0.038 and 0.507±0.059 for linear polymers HDPE and branched LDPE, respectively. These results reveal a common picture of hole creation above ${\mathit{T}}_{\mathit{g}}$ in polymers, however, the concrete value of β may be associated with the branching structure of macromolecules.