Ph.D. thesis (pdf) - dirac

30 What we learn from pressure experiments
T τ -line and isobaric fragility evaluated at a given relaxation time can be considered
as a function of pressure. Empirically, it is most often found that isobaric
fragility decreases with pressure, but it can also be increasing or virtually pressure
independent [Roland et al., 2005].
In addition to the isobaric fragility, it is also possible to define an isochoric fragility:
m ρ = ∂ log 10(τ)
∂ T τ /T
∣ (T = T τ ). (3.1.2)
ρ
The isochoric fragility is a measure of how much the temperature dependence of
the relaxation time departs from Arrhenius when the liquid is subjected to isochoric
cooling. Isochoric cooling is often performed in simulations which makes this
distinction particularly important when comparing experimental and simulation results.
Experimentally, it is difficult to perform isochoric cooling, but the isochoric
derivative is nonetheless a well defined quantity.
The two fragilities are straightforwardly related by the chain rule of differentiation:
m P = ∂ log 10(τ)
∂ T τ /T
= m ρ + ∂ log 10(τ)
∂ρ
∣ (T = T τ ) + ∂ log 10(τ)
∂ρ
ρ
∂ρ ∣
T
∂ T τ /T ∣ (T = T τ ) (3.1.3)
P ∣
∂ρ
∂ T τ /T ∣ (T = T τ ) (3.1.4)
P
∣
T
when both are evaluated at the same thermodynamic state point, e.g. at a given
pressure P 1 defining (T τ (P 1 ), ρ(P 1 , T τ (P 1 ))). Expressed in this way the effects leading
to the slowing down when cooling along an isobar is separated in two contributions:
(i) the slowing down due to the decrease of temperature itself, and (ii) the contribution
from the increase of density which follows as a consequence of decreasing
temperature [Ferrer et al., 1998].
Equation 3.1.3 can be rewritten to
m P = m ρ (1 − α P /α τ ) (3.1.5)
where α P is the isobaric expansivity α P = −1 ∂ρ
ρ ∂T
∣ while α τ = −1 ∂ρ
P
ρ ∂T
∣ is the
τ
isochronic expansivity; that is a measure of volume changes as a function of temperature
along an isochrone (a line where the alpha relaxation time is constant) [Ferrer
et al., 1998]. We want to stress that equation 3.1.5 is exact. It only builds on the
definitions introduced and some standard differential algebra 1 .
1 The equivalence between equation 3.1.3 and 3.1.5 can be shown from the relation
˛ ∂T τ /T ˛ ∂ρ
∂ρ ∂log τ
˛˛˛T=-1.
∂ log 10 (τ)
∂ T τ /T
˛ρ
˛τ