Vogel-Fulcher law of glass viscosity: A new approach

Abstract

Starting with an expression, due originally to Einstein, for the shear viscosity \textit{$η$}(\textit{$δϕ$}) of a liquid having a small fraction \textit{$δϕ$}by volume of solid particulate matter suspended in it at random, we derive an effective-medium viscosity \textit{$η$}(\textit{$ϕ$}) for arbitrary \textit{$ϕ$} which is precisely of the Vogel-Fulcher form. An essential point of the derivation is the incorporation of the excluded-volume effect at each turn of the iteration \textit{$ϕ$}$_{n + 1 =}$\textit{$ϕ$}$_{n}$\textit{+$δϕ$}. The model is frankly mechanical, but applicable directly to soft matter like a dense suspension of microspheres in a liquid as function of the number density. Extension to a glass forming supercooled liquid is plausible inasmuch as the latter may be modelled statistically as a mixture of rigid, solid-like regions (\textit{$ϕ$}) and floppy, liquid-like regions (1-\textit{$ϕ$}), for \textit{$ϕ$} increasing monotonically with supercooling.