Ph.D. thesis (pdf) - dirac

7.5. Temperature dependence 133
7.5 Temperature dependence
The figure 7.13 shows the 〈u 2 〉 T /〈u 2 〉 Tg as a function of T/T g . Hence the figure
illustrates the relative change in the total 〈u 2 〉 as a function of the relative change
in T. The 〈u 2 〉 value of the very fragile DHIQ rises most dramatically, the 〈u 2 〉 of
glycerol the least, while the three remaining liquids, which all have similar intermediate
fragilities fall in between. The five systems studied hence follow the general
trend that more fragile liquids have more temperature dependent mean square displacement
above T g [Ngai, 2004].
/ (T g
)
3
2.5
2
1.5
DHIQ
cumene
m−Toluidine
DBP
Glycerol
1
0.9 1 1.1 1.2 1.3
T/T g
Figure 7.13: 〈u 2 〉 scaled to 〈u 2 〉 Tg for 5 different liquids. The temperature is scaled
to T g .
The elastic model moreover makes a quantitative prediction regarding the relation
between the temperature dependence of 〈u 2 〉 and that of the alpha relaxation time.
The elastic model leading to equation 7.3.1 is based on
E(ρ, T)
k B T
= Cρ−(2/3)
〈u 2 〉(ρ, T)
(7.5.1)
(assuming that a 2 ∝ ρ −2/3 ). Recalling the definition of the Olsen fragility index
(equation 2.2.5) it follows that the model predicts (a special case of equation 3.4.3)
dlog E(ρ, T)
I P = − dlog T ∣ (7.5.2)
P
= − dlog T
dlog T + ∂ log〈u2 〉
∂ log T ∣ + 2 ∂ log ρ
P
3 ∂ log T ∣ (7.5.3)
P
= −1 + ∂ log〈u2 〉
∂ log T ∣ − 2
P
3 Tα P. (7.5.4)