2.6. Arrhenius and Vogel-Fulcher-Tamman Models for the Concentration Dependence of Viscosity and Relaxation Time

Viscosity of liquids approaching the glass transition depends strongly on concentration and/or temperature and it usually increases with increasing concentration or decreasing temperature, this behaviour microscopically corresponding to a slowing down of the dynamics. When viscosity η shows a low sensitivity to small changes of the control parameters (Cw, T…) it follows the well known exponential Arrhenius-like behaviour [69] that in the case of concentration can be written as:

where η0 is the viscosity in the limit of Cw = 0 and A controls the growth of the function. Conversely, when η shows a very steep increase with respect to small changes in concentration, viscosity is well described by the Vogel-Fulcher-Tammann (VFT) model [70] identified by an exponential with three free parameters:

where η0 is the viscosity in the limit of Cw = 0, A is the growth parameter and Cw0 is the critical concentration that signs the divergence of η. Both behaviours have been observed in a wide range of systems [70,71] including soft materials [14,19,58,72,73,74,75] and are precursor of a glass transition that happens at the glass transition temperature Tg or concentration Cwg. In the renowned Angell classification [70,75], by cooling the system up to Tg the typical system time scale becomes of the order of 102s and it is accompanied by a simultaneous increase of the viscosity that, around the glass transition, is of the order of 1012Pa·s [70]. Systems that follow the Arrhenius law are denoted as “strong glasses” and those that follow the VFT model are defined as ”fragile glasses” [76]. Soft and compressible particles, such as microgels, undergo the glass transition when their concentration approaches the critical value. These systems, depending on softness, can show an Arrhenius or a Vogel-Fulcher-Tamman behaviour resembling that of molecular glasses [14,19,58,75].