Ph.D. thesis (pdf) - dirac

2.1. The glass transition 9
Cooling at very high rates is called quenching. The glasses formed by quenching have
larger specific volume because their properties are frozen in at a higher temperature
and a corresponding higher volume. The term T g is traditionally used about the
temperature at which the liquid falls out of equilibrium when cooled at standard
experimental rates [Ediger et al., 1996]. This in practice happens at the temperature
where the viscosity is 10 12 Pas ∼ 10 13 Pas and the alpha relaxation time (τ α ) is of
the order 100 s ∼ 1000 s. The criterion τ α = 100 s is often used as a definition of
the glass transition temperature.
The traditional route to glass-formation is to cool the liquid at constant atmospheric
pressure. However, the characteristic alpha relaxation time also increases
when pressure is increased along an isotherm. This leads to a freezing in of the
structural relaxation at a given pressure P g , where the relaxation time has reached
100 s ∼ 1000 s. The effects of pressure and temperature on the viscous slowing down
can be considered jointly by describing the alpha relaxation time as a function of the
two: τ α (P, T). Based on this function it is possible to determine lines of constant
alpha relaxation time in the parameter space defined by pressure and temperature.
We shall refer to lines of constant relaxation time as isochrones and consider the
T g (P) line as a special case of an isochrone.
It has been suggested that the viscous slowing down observed at atmospheric pressure
is due to the decrease of the specific volume which follows from cooling [Cohen
and Turnbull, 1959]. However, measurements of the relaxation time as a function
of temperature and pressure have clearly shown that volume alone does not control
the relaxation time. One way to illustrate this is by showing that the isochrones are
not parallel to the isochores in the T − P diagram.
The other extreme would be a situation where the relaxation time is only temperature
dependent. The simplest model of the temperature dependence would be
an activated behavior, where the viscosity or relaxation time is controlled by some
temperature independent activation energy (E a , measured in units of temperature).
This would lead to an Arrhenius temperature dependence:
η = η p exp
(
Ea
T
)
and
τ = τ 0 exp
(
Ea
T
)
, (2.1.1)
where η p and τ 0 are the high temperature limits of the viscosity and the alpha
relaxation time respectively. Arrhenius behavior is actually (almost) followed by
some systems (see below), but this is not the general case. The dependence on
temperature is usually super-Arrhenius, i.e. stronger than the Arrhenius form. It
is possible to keep the notion of an activated behavior by allowing the activation