Viscosity of suspensions and glass: turning power-law divergence into essential singularity

Abstract

Starting with an expression, due originally to Einstein, for the shear viscosity η(δΦ) of a liquid having a small fraction δΦ by volume of solid particulate matter suspended in it at random, an effective-medium viscosity η(Φ) for arbitrary Φ is derived, 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 Φ_n+1=Φ_n+δΦ. The model is frankly mechanical, but applicable directly to soft matter like a dense suspension of microspheres in a liquid as a 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 (Φ) and floppy, liquid-like regions (1-Φ), for Φ increasing monotonically with supercooling.