Abstract
The three equations involved in the time-temperature superposition (TTS) of a polymer, i.e., Williams–Landel–Ferry (WLF), Vogel–Fulcher–Tammann–Hesse (VFTH) and the Arrhenius equation, were re-examined, and the mathematical equivalence of the WLF form to the Arrhenius form was revealed. As a result, a developed WLF (DWLF) equation was established to describe the temperature dependence of relaxation property for the polymer ranging from secondary relaxation to terminal flow, and its necessary criteria for universal application were proposed. TTS results of viscoelastic behavior for different polymers including isotactic polypropylene (iPP), high density polyethylene (HDPE), low density polyethylene (LDPE) and ethylene-propylene rubber (EPR) were well achieved by the DWLF equation at high temperatures. Through investigating the phase-separation behavior of poly(methyl methacrylate)/poly(styrene-co-maleic anhydride) (PMMA/SMA) and iPP/EPR blends, it was found that the DWLF equation can describe the phase separation behavior of the amorphous/amorphous blend well, while the nucleation process leads to a smaller shift factor for the crystalline/amorphous blend in the melting temperature region. Either the TTS of polystyrene (PS) and PMMA or the secondary relaxations of PMMA and polyvinyl chloride (PVC) confirmed that the Arrhenius equation can be valid only in the high temperature region and invalid in the vicinity of glass transition due to the strong dependence of apparent activation energy on temperature; while the DWLF equation can be employed in the whole temperature region including secondary relaxation and from glass transition to terminal relaxation. The theoretical explanation for the universal application of the DWLF equation was also revealed through discussing the influences of free volume and chemical structure on the activation energy of polymer relaxations.
Highlights
As one of the most important issues in polymer science, the time-temperature superposition (TTS)principle is significant and widely used because it describes the time-temperature equivalence for polymer relaxations ranging from secondary relaxations, segmental relaxation to terminal flow, which extends the test range both in experiments and in practice
According to the fact that the temperature dependence of segment diffusion and diffusion-controlled reaction in polymer systems can be described by the WLF equation [8,11], we have put forward an approach to obtain the equilibrium spinodal temperature by introducing a WLF-like equation to analyze the data obtained by small-angle laser light scattering (SALLS) during phase-separation of poly(methyl methacrylate)/poly(styrene-co-acrylonitrile) (PMMA/SAN) [13]
When the temperature is near the glass transition, the molecular relaxation rate is strongly limited by the large energy barrier, which is required for redistributing the free volume [47,48]
Summary
As one of the most important issues in polymer science, the time-temperature superposition (TTS). Obtained a good fit for the temperature dependence of viscosity for different liquids over a wide temperature range Those results show an expectation that exploring a universal equation to describe the relaxations above Tg may be possible based on the same primitive friction factor. Considering the relaxation similarity of motion units at different levels for polymers, we try to reveal the internal relationship among the three equations and find a universal form to describe the relaxations over a wide temperature range from both points of experiment and theory, since it is significant for polymer theory, and helpful for the experimental methods and practice. Through discussing the activation energy of polymer relaxations, the theoretical rationality for the DWLF equation was revealed
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