Glass Transition Seen through Asymptotic Expansions

Abstract

Soft glassy materials exhibit the so-called glassy transition, which means that the behavior of the model at a low shear rate changes when a certain parameter (which we call the glass parameter) crosses a critical value. This behavior goes from a Newtonian behavior to a Herschel–Bulkley behavior through a power-law-type behavior at the transition point. In a previous paper we rigorously proved that the Hébraud–Lequeux model, a Fokker–Planck-like description of soft glassy material, exhibits such a glass transition. But the method we used was very specific to the one-dimensional setting of the model, and as a preparation for generalizing this model to take into account multidimensional situations, we look for another technique to study the glass transition of this type of model. In this paper we shall use matched asymptotic expansions for such a study. The difficulties encountered when using asymptotic expansions for the Hébraud–Lequeux model are that multiple ansaetze have to be used, even though the initial model is unique, due to the glass transition. We shall delineate the various regimes and give a rigorous justification of the expansion by means of an implicit function argument. The use of a two parameter expansion plays a crucial role in elucidating the reasons for the scalings which occur.