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3.2. Empirical scaling law and some consequences 31
Equation 3.1.5 shows that the difference between m P and m ρ is determined by the
ratio of two expansivities, α τ and α P . However, the difference is not determined
by thermodynamics alone because α τ contains dynamical information as well, since
it is necessary to know the slope of the isochrone (e.g. the glass transition line) in
order to evaluate it.
Turning now to the phenomenology, it is well known that α P is positive 2 ; α τ on the
other hand is negative because density increases as with increasing temperature when
moving along an isochrone (see figure 3.1 a). By inserting these simple empirical
facts in equation 3.1.5 can be seen that the isobaric fragility is larger than the
isochoric fragility.
3.2 Empirical scaling law and some consequences
Within the last decade a substantial amount of relaxation time and viscosity data
has been collected at different temperatures and pressures/densities, mainly by the
use of dielectric spectroscopy. On the basis of the existing data it is relatively well
established that the temperature and density dependence of the relaxation times
can be expressed as first suggested by Alba-Simionesco et al. [2002], as
( ) e(ρ)
τ(ρ, T) = F . (3.2.1)
T
The result is empirical and has been supported by the work of several groups for a
variety of glass-forming liquids and polymers [Alba-Simionesco et al., 2002; Tarjus
et al., 2004 a; Casalini and Roland, 2004; Roland et al., 2005; Dreyfus et al., 2004;
Reiser et al., 2005; Floudas et al., 2006]. See also chapter 5 in this work.
3.2.1 The result and its history
The scaling can also be expressed in terms of the activation energy defined in equation
2.1.2. In fact is was first proposed in its general form from the idea of reducing
the influence of density on the slowing down to a single density dependent activation
energy scale [Alba-Simionesco et al., 2002; Alba-Simionesco and Tarjus, 2006]:
( )
E(ρ, T) T
E ∞ (ρ) = Φ . (3.2.2)
E ∞ (ρ)
2 Except for tetrahedral systems at certain temperatures, eg. water below 4 ◦ C.