Twinkling fractal theory of the glass transition


  • Richard P. Wool

    Corresponding author
    1. Department of Chemical Engineering, Center for Composite Materials, University of Delaware, Newark, Delaware 19716-3144
    • Department of Chemical Engineering, Center for Composite Materials, University of Delaware, Newark, Delaware 19716-3144
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  • Presented at the American Physical Society, New Orleans, March 10, 2008.


In this paper we propose a solution to an unsolved problem in solid state physics, namely, the nature and structure of the glass transition in amorphous materials. The development of dynamic percolating fractal structures near Tg is the main element of the Twinkling Fractal Theory (TFT) presented herein and the percolating fractal twinkles with a frequency spectrum F(ω) ∼ ωdf–1 exp −|ΔE|/kT as solid and liquid clusters interchange with frequency ω. The Orbach vibrational density of states for a fractal is g(ω) ∼ ωdf–1, where df = 4/3 and the temperature dependent activation energy behaves as ΔE ∼ (T2Tmath image). The key concept of the TFT derives from the Boltzmann population of excited states in the anharmonic intermolecular potential between atoms, coupled with percolating solid fractal structures near Tg. The twinkling fractal spectrum F(ω) at Tg predicts the correct dynamic heterogeneity behavior via the spatio-temporal thermal fluctuation autocorrelation relaxation function C(t). This function behaves as C(t) ∼ t−1/3 (short times), C(t) ∼ t−4/3 (long times) and C(t) ∼ t−2 (ω < ωc), which were found to be in excellent agreement with published nanoscale AFM dielectric force fluctuation experiments on a glassy polymer near Tg. Using the Morse potential, the TFT predicts that Tg = 2Do/9k, where Do is the interatomic bonding energy ∼ 2–5 kcal/mol and is comparable to the heat of fusion ΔHf. Because anharmonicity controls both the thermal expansion coefficient αL and Tg, the TFT uniquely predicts that αL×Tg ≈ 0.03, which is found to be universal for a broad range of glassy materials from Pyrex to polymers to glycerol. Below Tg, the glassy structure attains a frustrated nonequilibrium state by getting constrained on the fractal structure and the thermal expansion in the glass is reduced by the percolation threshold pc as αgpcαL. The change in heat capacity ΔCp = CpLCpg at Tg was found to be related to the change in dimensionality from Df to 3 in the Debye approximation as the ratio CpL/Cpg = 3/Df, where Df is the fractal dimension of the glass. For polymers, the TFT describes the molecular weight dependence of Tg, the role of crosslinks on Tg, the Flory-Fox rule of mixtures and the WLF relation for the time-temperature shift factor aT, which are traditionally viewed in terms of Free-Volume theory. The TFT offers new insight into the behavior of nano-confined glassy materials and the dynamics of physical aging. It also predicts the relation between the melting point Tm and Tg as Tm/Tg = 1/[1−pc] ≈ 2. The TFT is universal to all glass forming liquids. © 2008 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 46: 2765–2778, 2008