Ph.D. thesis (pdf) - dirac

130 Mean squared displacement
cumene fall closer to the much less fragile glycerol than it does to m-toluidine, which
has fragility similar to that of cumene.
As an alternative figure 7.9 shows the absolute values of mean square displacement
for the five liquids in one plot. The temperature is again scaled to T g , but the 〈u 2 〉’s
are given in absolute units of Å 2 . This plot is the same type of plot as the ones
presented in figure 4 and figure 5 of [Ngai, 2004]. It is this type of plot which is used
to argue that the absolute values of 〈u 2 〉 at T g are larger for liquids with larger n
(smaller β KWW larger fragility in the frame of Ngai’s coupling model). In figure 7.9
we see that least fragile liquid, glycerol does indeed have the lowest absolute value.
However, the extremely fragile DHIQ falls just between the liquids with much lower
fragility, at odds with the results reported by Ngai [2004].
[A 2 ]
1
0.8
0.6
0.4
DHIQ
cumene
m−Toluidine
DBP
Glycerol
0.2
0
0 0.2 0.4 0.6 0.8 1 1.2
T/T g
Figure 7.9: 〈u 2 〉 as a function of temperature for 5 different liquids. The temperature
is scaled to T g .
7.4.1 Pressure dependence
The value of 〈u 2 〉/a 2 at T g does not appear to be universal, when considering the
5 liquids studied here. This breakdown of the Lindemann criterion can be ad hoc
explained by allowing C in equation 7.3.1 to be material dependent. The Lindemann
prediction becomes weaker, but can still be scrutinized by looking at the pressure
dependent T g of a given system. The “pressure dependent” Lindemann criterion
following from equation 7.3.1 says that 〈u 2 〉/a 2 should be constant on an isochrone,
particularly T g (P) (This is a special case of equation 3.4.2). The change in density
will lead to a change in 〈u 2 〉 but also to a change in a 2 . The simplest assumption for
the density dependence is to assume that it follows the change in density; a ∝ ρ −1/3
[Dyre, 2006].
Figure 7.10 shows the temperature dependence of the mean square displacement in