M2 Viscosity: Vogel-Fulcher-Tammann (VFT) Law

Topic

The progressive collapse of this glass's excess free volume—as the temperature drops toward the glass transition (which is defined by a different model within this framework)—is what manifests in this model as the divergence of the glass's viscosity. Unlike a pure Arrhenius dependence—where viscosity would merely increase exponentially as temperature falls, without ever diverging at a finite temperature—the Vogel-Fulcher-Tammann equation describes a viscosity that diverges (becoming infinite) at a specific, finite divergence temperature. Because the excess free volume continues to collapse as the temperature drops, the Vogel-Fulcher-Tammann equation predicts that viscosity increases at an accelerating rate until it diverges at that finite temperature, rather than diverging only at absolute zero, as a pure Arrhenius dependence would imply.

Angell's fragility index classifies materials based on the extent to which their viscosity deviates from pure Arrhenius behavior as the temperature approaches that finite divergence temperature: a "strong" glass (in the kinetic sense used by this index) is one whose viscosity deviates little from pure Arrhenius behavior, whereas a "fragile" glass (in that same kinetic sense) is one whose viscosity deviates significantly, rising much more abruptly as the temperature nears the finite divergence temperature. Consequently, a higher Angell fragility index indicates a more pronounced deviation from pure Arrhenius behavior.

This finite viscosity divergence temperature is empirically linked to the glass's glass transition temperature; the two temperatures lie relatively close to one another because, from different perspectives, they both mark the same collapse in molecular mobility as the glass cools. The fact that this glass has a higher glass transition temperature implies—based on this same empirical relationship—that the finite temperature at which viscosity diverges is also higher.

Consequently, this model—characterized by the viscosity divergence mentioned at the outset—serves as the central equation distinguishing this glass from a crystalline solid; the latter’s viscosity does not diverge in this continuous manner but instead drops abruptly at its melting temperature. Because the Vogel-Fulcher-Tammann equation describes the glass's viscosity across its entire temperature range, two other models in the set—which measure viscosity using a rotational viscometer in the melt range and a parallel-plate viscometer in the annealing range—can fit their parameters using these combined experimental data; furthermore, since this same equation determines the glass's standard viscosity points, a distinct model in the set can utilize the derived relaxation time near the annealing temperature.

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