M3 Structural Relaxation and Fictitious Temperature

Topic

The fact that this glass passes through the glass transition—a phenomenon already established in another model of this set—during cooling, and that its structure becomes frozen out of equilibrium upon crossing it, is what characterizes a "fictive temperature" distinct from the glass's actual temperature in this model: the fictive temperature is the temperature at which the glass's frozen structure would actually be in equilibrium, even though the glass's actual temperature—once cooled below the glass transition—is different. A faster cooling rate through the glass transition causes the structure to freeze further from equilibrium; consequently, the resulting fictive temperature is higher and further removed from the glass's actual temperature.

Once the glass is below the glass transition, this fictive temperature does not remain fixed; instead, it slowly relaxes toward the glass's actual temperature, as described by the Tool-Narayanaswamy-Moynihan model. If the glass remains at a specific temperature below the glass transition for a longer period, the model predicts that its fictive temperature will progressively approach the actual temperature, though never instantaneously.

However, this relaxation of the fictive temperature does not follow the simple form of a single exponential decay over time; rather, it takes the more complex form described by the KWW (Kohlrausch-Williams-Watts) function. Instead of proceeding at a single, uniform rate, this relaxation combines multiple, superimposed processes—some faster, some slower—causing it to extend over a much longer period than a simple exponential decay would predict. Because the KWW function describes a relaxation that is more "stretched" in time, the glass's fictive temperature takes longer to converge toward its actual temperature than it would under simple exponential relaxation. For this reason, the model—characterized by its out-of-equilibrium structure and the associated fictive temperature—describes a Narayanaswamy relaxation time that is both non-linear and non-Arrhenius: non-linear, because the relaxation time depends on the fictive temperature itself rather than solely on the glass's actual temperature (meaning two samples at the same actual temperature but with different thermal histories relax at different rates); and non-Arrhenius, because the relaxation time does not follow the simple exponential temperature dependence described by the Arrhenius law. The fact that this structural relaxation—with its fictive temperature and non-linear relaxation time—depends entirely on the existence of an out-of-equilibrium structure means the model has no analogue in crystalline solids; the latter possess an equilibrium structure that requires no fictive temperature to distinguish it from the actual temperature. Furthermore, this allows another model within the same framework to utilize this same Narayanaswamy relaxation time—near the annealing temperature—to determine the time required to anneal the glass.

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