Correlation between macroscopic surface tension (γ) and o-Ps lifetime (τo-Ps) for PDMS (Ref. 19), PIB (Ref. 21), and PB (Ref. 21). The dotted brown curve is a fit to the points for the three polymers above their respective knee temperatures (T > Tk). Glass transition temperatures (Tg) are indicated. So are the ‘knee’ temperatures (Tk) above which o-Ps bubbles appear to be formed. The two thin black curves define the range of values for the τo-Ps vs. γ correlation observed for molecular liquids by Tao (Ref. 3) (Eq. (1)).
Simple, schematic illustration of the expansion of a free volume cavity. In this simple 2-dimensional sketch of the 3-dimensional structure of cavity and molecules, the light grey circles (spheres) represent monomers or segments of the polymer chains, while the small, darker circle represents the Ps atom localized in the cavity. The cavity may expand from the size of a free volume cavity (of radius RFV) to the equilibrium bubble (of radius Rbubble), as a result of the Ps outward pressure (PPs – dotted arrows). The pressure due to the surface tension (Pγ – empty arrows) counteracts expansion. Filled arrows indicate direction of moving molecular segments between which viscous friction occurs, causing dissipation of energy.
Cavity sizes Vo-Ps (green triangles) in poly(isobutylene) derived from o-Ps lifetimes measured by Zrubcova et al. (Ref. 21) in the temperature range between 80 and 460 K using Eqs. (4). The dash-dotted line marked VFV shows free volume cavity sizes linearly extrapolated (Eq. (7)) from values in the range Tg < T < 270 K. The brown dotted curve identified as Vequilibrium shows the hypothetical equilibrium bubble size in the whole temperature range, as derived from the dotted curve in Fig. 1. The glass transition temperature (Tg) and knee temperature (Tk) are indicated. The latter is associated with the onset of equilibrium o-Ps bubbles.
Comparison of macroscopic (✝, ηmacro) and estimated nanoscopic viscosity (□, ηo-Ps) of PIB. The macroscopic viscosity was calculated from a relation by Fox and Flory (Ref. 40). The lengths of the vertical bars used as symbols mark the variation of the calculated viscosity for different molecular weights (Ref. 41). The nanoscopic viscosity was estimated using the Navier-Stokes equation (see text). Inserted in top right corner is a log-log plot of the estimated correlation between ηmacro and ηo-Ps which has been approximated by a straight line (Eq. (8)).
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