Details

Snapshot 1: viscosity versus plots where corresponds to viscosity of Pa s, generated with the WLF model having the same parameters used to generate the thumbnail

Snapshot 2: viscosity versus plots where corresponds to viscosity of Pa s, generated with the WLF model with different parameters than those used to generate the thumbnail and Snapshot 1

Snapshot 3: viscosity versus plots where is defined by the viscosity of Pa s, generated with the Arrhenius model

Snapshot 4: viscosity versus plots where corresponds to viscosity of Pa s, generated with the Arrhenius model with different parameters than those used to generate Snapshot 3

The object of this Demonstration is to show that the convergence of such curves occurs regardless of the chosen viscosity-temperature model, the magnitudes of its parameters, the assigned to the materials and the chosen viscosity level that defines it.

Perhaps the best-known viscosity-temperature model is the two-parameter WLF model [4], which for our purpose can be written in the form

,

where and are the viscosities at temperatures and respectively, both in , and , are constants. This model has been widely used to replace the single-parameter Arrhenius equation, which may only apply at higher temperatures. For our purpose, the Arrhenius equation can be written in the form

where the temperatures are in , is the "energy of activation" and is the universal gas constant in commensurate units. It can be shown that if the absolute temperatures in the Arrhenius equation, and , are replaced by and where is an adjustable constant (all in ), then the resulting expanded model becomes the same as the WLF model and thus also the VTF (or VFT) model, which has long been known as its equivalent [5].

For what follows we will use the original version of the Arrhenius model. While it is usually not appropriate for temperatures in the neighborhood of the glass transition, it can still show that the viscosity-temperature curve's convergence to a single point is independent of the chosen model.

To run the Demonstration, select the viscosity-temperature model, WLF or Arrhenius, with its setter bar and then choose the viscosity level that defines in the range from to Pa s. Then choose the characteristics of three hypothetical or actual different materials, 1, 2 and 3, whose curves will be shown in red, green and blue, respectively. Use their sliders to enter or vary their settings in , the and parameters of the WLF model, or the of the Arrhenius model.

As expected, regardless of the particular settings, the three curves will always converge to the point where and the viscosity is that which is used to define the glass transition temperature.

References

[1] Wikipedia. "Glass Transition." (Jun 6, 2019) en.wikipedia.org/wiki/Glass_transition.

[2] C. A. Angell, "Formation of Glasses from Liquids and Biopolymers," Science, 267(5206), 1995 pp. 1924–1935. www.jstor.org/stable/2886440.

[3] J. C. Mauro, Y. Yue, A. J. Ellison, P. K. Gupta and D. C. Allan, "Viscosity of Glass-Forming Liquids," Proceedings of the National Academy of Sciences, 106(47), 2009 pp. 19780–19784. doi:10.1073/pnas.0911705106.

[4] M. L. Williams, R. F. Landel and J. D. Ferry, "The Temperature Dependence of Relaxation Mechanisms in Amorphous Polymers and Other Glass-Forming Liquids," Journal of the American Chemical Society, 77(14), 1955 pp. 3701–3707. doi:10.1021/ja01619a008.

[5] M. Peleg, "Temperature-Viscosity Models Reassessed," Critical Reviews in Food Science and Nutrition, 58(15), 2018 pp. 2663–2672. doi:10.1080/10408398.2017.1325836.