Ph.D. thesis (pdf) - dirac

3.2. Empirical scaling law and some consequences 35
relaxation time:
m ρ = dlog 10(τ)
dT τ /T
∣ (T = T τ ) = F ′ dX
(X τ )
ρ
dT τ /T (T = T τ) = X τ F ′ (X τ ). (3.2.6)
The physical meaning is that temperature T τ changes as a function of pressure, but
the relaxation as a function of temperature will have the same degree of departure
from Arrhenius along different isochores. We illustrate this situation in figure 3.1.
The fact that the relaxation time τ is constant when X is constant means that the
isochronic expansion coefficient α τ is equal ) to the)
expansion coefficient at constant
)
X. Using this and the general result
= −1, it follows that
(
∂ρ
∂T
which inserted in equation 3.1.5 leads to
X
(
∂X
∂ρ
T
( ∂T
∂X
1 dlog e(ρ)
= −T
α τ dlog ρ , (3.2.7)
(
)
dlog e(ρ)
m P = m ρ 1 + α P T τ , (3.2.8)
dlog ρ
where m P and m ρ are again evaluated at a given relaxation time τ.
ρ
This expression illustrates that the relative effect of density on the slowing down
upon isobaric cooling, i.e., the second term in the parentheses, can be decomposed
into two parts: the temperature dependence of the density measured by T τ α P =
− ∂ log ρ
∣ (T = T τ ) , and the density dependence of the activation energy, which is
P
∂ log T
contained in
dlog e(ρ)
d log ρ .
Since m ρ is constant along an isochrone, it follows from equations 3.1.5 and 3.2.8
that the change in m P with increasing pressure is due to the change in α P /α τ =
α P T τ
dlog e(ρ)
d log ρ .
T τ increases with pressure, α P T τ (P) decreases, whereas
dlog e(ρ)
dlog ρ
= x is often to
a good approximation constant in the range of densities accessible 3 . The most
common behavior seen from the data compiled by Roland et al. [2005] is that the
isobaric fragility decreases or stays constant with pressure, with few exceptions. This
indicates that the decrease of α P T g (P) usually dominates over the other factors.
3 The DBP case at high density discussed in section 5.2.1 is one exception.