Vogel-fulcher Law of Glass Viscosity: a New Approach

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, we derive an effective-medium viscosity η(φ) for arbitrary 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 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 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. Extreme slow dynamics defines approach to the glassy state. At the macroscopic scale, it manifests as a rise of shear viscosity, typically by 15 orders of magnitude, as that state is reached through supercooling of the glass forming liquid. The Vogel-Fulcher (VF) law describes that growth of viscosity [1].This work derives the VF law [2]. A striking feature of the VF law is the essential singularity, rather than a power-law divergence, of the shear viscosity at a temperature T 0. The relaxation times, however, exceed the experimental time scale at what is identified as the glass transition temperature T g > T 0 , making thus the glass transition a kinetic crossover. This inverse exponential VF law is well known to hold for the fragile structural-glass forming liquids [1]. But, very significantly it is also obeyed by a broad class of soft-matter systems that exhibit the extreme slow dynamics [1]. This includes the purely mechanical systems, e.g., of weakly perturbed granular aggregates, where the degree of compaction and the perturbation strength, rather than mass density and temperature, are the relevant variables and the control parameter, and the underlying physics is that of jamming, or blocking, by rigid granular contacts [3-6]. And, similarly for the case of a dense suspension of microspheres [1]. Motivated by its ubiquity and universality, we have attempted a derivation of the VF law for a fluid-mechanical model of a liquid containing a volume fraction φ of solid 1