Mathematical models of glass formation

Abstract

This article presents a theory of glass formation based on the mathematical description of agglomeration at the microscopic level. This theoretical model displays a nonnegligible predictive power when applied to the description of the medium-range order in binary covalent glasses such as borates, binary or ternary chalcogenide glasses, and the best known ternary silicon oxide-based glass. The mathematical approach used consists in spotting first the most stable elementary configurations in glass, treated asbuilding blocks, then a careful counting of all types of clusters containing many such entities and evaluation of the number of different pathways leading to their formation. The probabilities of cluster forming are then computed taking into account the multiplicities (statistical factors) and the Boltzmann factors related to the different energetic costs of various agglomeration processes. The probabilities of characteristic features observed in a given glass (e.g., boroxol rings, edge-sharing or vertex-sharing doublets of rings) are then computed; finally, a system of nonlinear equations can be produced by requiring the time derivatives of these probabilities to vanish, which can be interpreted as thesaturation or thestationary regime of the agglomeration process attained at a relatively early stage of glass formation that also corresponds tominimum fluctuations. The most important result of this type of modeling is the correct prediction of the glass transition temperature dependence on modifier concentration.