Apparaît dans la collection : Combinatorics and Arithmetic for Physics : Special Days
Glasses are characterized by the absence of long-range order which
defines crystalline materials. However, they possess a rich and varied array of
short to medium range order, which originates from chemical bonding and related
interactions. whereas covalent systems (mostly chalcogenides like As-Se, Ge-As-
Se systems) or oxides (borate, boro-silicate and silicate glasses), have sparsely
packed, strongly bound network structures, like tetrahedral SiO2 units or B3O3
boroxol rings. These very different structures results in different physical proper-
ties and applications.
We present a simple mathematical model of glass transition based on the anal-
ysis of molecular agglomeration in overcooled liquids. The model uses the space
of probabilities of appearance of given local structures, and their slow time evo-
lution during annealing from a liquid melt. The evolution of probabilities is de-
scribed as action of an appropriate stochactis matrix. The glass transition is de-
fined as a fixed point resulting from the requirement of maximal homogeneity.
With simple assumptions concerning local configurations and their bonding en-
ergies, and with elementary combinatorics we are able to derive the dependence
of the glass transition temperature Tg on chemical composition in non-organic
covalent glasses. Numerous examples are shown to confirm the validity of the
stochastic agglomeration model.