We have reexamined the free-volume concept presented by Cohen and Turnbull on the basis of two microscopic quantities: the excess mean-square displacement $〈u^2{〉}_f$ and the total free volume $V_{\mathrm{PA},t},$ of poly- butadiene evaluated from the quasielastic neutron scattering and the positron annihilation lifetime spectroscopy (PALS) data, respectively. Comparing with the viscosity η we found two relations, $\eta ={\eta }_0\exp u_0^2/〈u^2{〉}_f$ and $\eta ={\eta }_0\exp V_{\mathrm{PA},0\mathrm{}}^{*}{/V}_{\mathrm{PA},t}={\eta }_0\exp V_{\mathrm{PA},0\mathrm{}}^{*}{/v}_{\mathrm{PA},f},$ where $u_0^2,V_{\mathrm{PA},0\mathrm{}}^{*}$ and $v_{\mathrm{PA},0\mathrm{}}^{*}$ are the critical values for the mean-square displacement, the total PALS free volume, and the PALS free volume per molecule, respectively, and further $v_{\mathrm{PA},0\mathrm{}}^{*}{=V}_{\mathrm{PA},0\mathrm{}}^{*}/N,$ N being the total number of molecules or segments. On the basis of these relations, we discuss the microscopic basis of the free-volume theory. The experimentally evaluated critical values $u_0^2$ and $v_{\mathrm{PA},0\mathrm{}}^{*}$ are much larger than the average values of $〈u^2{〉}_f$ and $v_{\mathrm{PA},f}$ calculated from the distributions. This has been explained from the low probability of escaping motions from a molecular cage. The free volume per monomer and the free-volume fraction were calculated from the excess mean-square displacement $〈u^2{〉}_f.$ The former was compared with the free-volume hole obtained by PALS, suggesting that 22 monomers are required for one PALS free-volume hole. The free-volume fraction obtained from the excess mean-square displacement was found to be 6.4% at 250 K, which is in reasonable agreement with that evaluated from the rheological data (9.0%).

• Received 8 September 1998

DOI:https://doi.org/10.1103/PhysRevE.60.1906