Twinkling fractal theory of the glass transition: Rate dependence and time–temperature superposition

Abstract

AbstractThe twinkling fractal theory (TFT) of the glass transition temperature Tg provides a new method of analyzing rate effects and time–temperature superposition in amorphous materials. The rate dependence of Tg was examined in the light of new experimental and theoretical evidence for the nature of the dynamic heterogeneity near Tg. As Tg is approached from above, dynamic solid fractal clusters begin to form and eventually percolate rigidity at Tg. The percolation cluster is a solid fractal and to the observer, appears to “twinkle” as solid and liquid clusters interchange in dynamic equilibrium with a vibrational density of states g(ω) ∼ ω. The solid‐to‐liquid twinkling frequencies ωTF are controlled by the Boltzmann population of intermolecular oscillators in excited energy levels of their anharmonic potential energy functions U(x) such that ωTF = ω exp −B(T*2 − T2)/kT in which T* ≈ 1.2Tg. An oscillator changes from a solid to a liquid when a thermal fluctuation causes it to expand beyond its inflection point in the anharmonic potential. This leads to a continuous solid fraction Ps near Tg given by PS ≈ 1−[(1 − pc) T/Tg] where pc ≈ 1/2 is the rigidity percolation threshold. Since g(ω) is continuous from very low to very high frequencies, the complex twinkling dynamics existing near Tg produces a continuous relaxation spectrum with many different length scales and times associated with the fractal clusters. The twinkling frequencies control the kinetics of Tg such that for a given observation time t when the rate γ > 1/t, only those parts of the twinkling spectrum with ω > γ can contribute to relaxation or percolation upto time t. The most important results in this article are as follows: The TFT describes the rate dependence of Tg, both for DSC thermal heating/cooling rates and DMA frequencies as the classic Tg − lnγ law as Tg(γ) = Tgo + (k/2B) ln γ/γo in which the constant B = 0.3 cal/mol K2. The constant B appears quite universal for the 17 thermoset polymers investigated in this study and 18 linear polymers investigated by others. Many other amorphous metal and ceramic glass materials exhibited the same rate law but required a new B value approximately half that for polymers. The same B = 0.3 value was also used to successfully describe the TTS shift factors using the twinkling fractal frequencies ωTF = ωexp −B(T*2 − T2)/kT, as ln aT(TFT) = exp B(TR2 − T2)/kT, which gave comparable results with the classical WLF equation, log aT = [−C1(T − TR)]/[C2 + (T − TR)]. The advantage of the TFT over the WLF is that C1 and C2 are not universal constants and must be determined for every material, whereas the TFT uses one known constant B which appears to be the same for all polymers. The TFT has also been found to describe the strong and fragile nature of the viscosity behavior of liquids and the rate and temperature dependence of the yield stress in polymers. © 2009 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 47: 2578–2590, 2009