Vogel-Fulcher law of glass viscosity: A new approach

TL;DR

This paper derives a Vogel-Fulcher form for glass viscosity starting from Einstein's shear viscosity expression, incorporating excluded-volume effects, and applies it to dense suspensions and supercooled liquids.

Contribution

It introduces a new effective-medium model for viscosity that naturally yields Vogel-Fulcher behavior, extending Einstein's approach with excluded-volume effects.

Findings

Derivation of Vogel-Fulcher viscosity form from microscopic considerations

Model applicability to dense suspensions and supercooled liquids

Inclusion of excluded-volume effects in viscosity modeling

Abstract

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