Twinkling Fractal Theory of the Glass Transition.

A new approach to the glass transition temperature T$_{g}$ considers the interaction of particles with an anharmonic potential U(x), and Boltzmann population $\phi $(x) $\sim $ exp --U(x)/kT. As T$_{g}$ is approached from above, solid clusters of atoms form and percolate at T$_{g}$. However, the solid percolation cluster is in dynamic equilibrium with its surrounding liquid and ``\textit{twinkles}'' as solid and liquid atoms interchange. The \textit{twinkling} frequency F($\omega )$ is related to the vibrational density of states G($\omega )$ $\sim \quad \omega ^{df}$ and the energy difference $\Delta $U $\sim $ (T$^{2}$-T$_{g}^{2})$ via F($\omega ) \quad \sim $ G($\omega )$ exp -$\Delta $U/kT, where d$_{f}$ = 4/3 is the fracton dimension. F($\omega )$ controls the rate dependence of T$_{g}$, physical aging, yield stress, heat capacity C$_{p}$, T$_{g}$ of thin films, etc. When T $<$ T$_{g}$, the non-equilibrium volume development $\Delta $V, is determined by the fractal structure at T$_{g}$.$_{ }$The thermal expansion coefficients in the liquid and glass are related via $\alpha _{g}$ = p$_{c}\alpha _{L}$. For a Morse potential U(x) = D$_{o}$[1-exp $a$x]$^{2}$, we predict that T$_{g}$ = 2D$_{o}$/9k, and $\alpha _{L}$ = 3k/[4D$_{o}$aR$_{o}$]. For atoms with R$_{o}\approx $ 3 {\AA}, bond energy D$_{o} \quad \approx $ 2-10 kcal/mol and anharmonicity factor $a\approx $2/{\AA}, we obtain $\alpha _{L}$ T$_{g} \quad \approx $ 0.03, and modulus E $\sim $ 1/$\alpha _{L}$, which were found for a broad range of polymers. The yield stress $\sigma _{y}$ is determined by the onset of the twinkling fractal state as $\sigma _{y}$ = {\{}0.16 E [p$_{s}$-p$_{c}$] D$_{o}$/V$_{m}${\}}$^{1/2}$ where V$_{m}$ is the molar volume.