Twinkling Fractal Analysis of Confinement Effects on the Glass Transition of Thin Films

The Twinkling Fractal Theory (TFT) of the Glass Transition has recently been verified experimentally [J.F. Stanzione, et al., ``Observing the twinkling fractal nature of the glass transition'', J. Non-Cryst. Solids (2010), doi:10.1016/j.jnoncrysol.2010.06.041] Here we apply the TFT to understand nanoconfinement effects on T$_{g}$ for amorphous thin films of thickness h with free and adhered surfaces. The TFT states that T$_{g}$ occurs when the dynamic clusters percolate rigidity at the rate of testing $\gamma $. The lifetime $\tau $ of these fractal clusters of size R behaves as $\tau \sim $R$^{\delta }$ exp $\Delta $E/kT, where $\delta $=D$_{f}$/d$_{f}$ in which D$_{f}$ is the fractal dimension and d$_{f}$=4/3 is the fracton dimension for the vibrational density of states g($\omega )\sim \omega ^{df}$. The activation energy $\Delta $E = $\beta $[T$^{\ast 2}$-T$_{g}^{2}$] in which $\beta \quad \approx $ 0.3 cal/mol $^{o}$K$^{2}$ and T*$\approx $1.2T$_{g}$. In confined spaces, only clusters of size R$<$h can exist and these have a very fast relaxation time compared to the bulk. Thus, for free surfaces, T$_{g}$ must be dropped at that test rate to percolate rigidity and we obtain the familiar expression T$_{g}$(h)/T$_{g\infty } \quad \approx $ [1-(B/h)$^{\delta }$] where $\delta \approx $1.8 when D$_{f} \quad \approx $2.5 and B is a known constant. For thin films adhered to solid substrates, T$_{g}$ increases in accord with the adhesion energy$\Delta $A as $\Delta $E$\to \Delta $E+$\Delta $A and the adhered cluster lifetime increases. As the rate of testing $\gamma $ increases, the confinement effects diminish as T$_{g}$ increases in accord with T$_{g}(\gamma )$ = T$_{go}$ + (k/2$\beta )$ ln $\gamma $/$\gamma _{o}$.