Ph.D. thesis (pdf) - dirac

128 Mean squared displacement
a central question for understanding the glass transition phenomenon [Angell et al.,
2000].
Several more specific relations between the temperature dependence of the alpha
relaxation time and the mean square displacement have been proposed. It is common
to these relations that they associate a larger mean square displacement to a shorter
alpha relaxation time, and hence expect the change of the mean square displacement
just above T g to be more dramatic the more fragile the liquid is. The physical
pictures and starting points vary, but the conclusions can essentially be condensed
to two different hypothesis.
One view is, that the activation energy related to the alpha relaxation time, should
be proportional to T and inversely proportional to the total temperature dependent
T
mean square displacement, E(ρ, T) ∝ , where a is a characteristic distance
a 2 〈u 2 〉(T)
between the relaxing entity. This yields:
( ) Ca
2
τ(T) = τ 0 exp
〈u 2 〉(T)
(7.3.1)
where C is a constant. This view point has mainly been based on so called elastic
models [Dyre and Olsen, 2004], (for a review see [Dyre, 2006]), but it has also been
proposed by Starr et al [Starr et al., 2002] based on a Voronoi volume analysis and
computer simulations.
The other view is that it is a non-harmonic part of 〈u 2 〉 that should be considered
instead of the total 〈u 2 〉. That is E ∝
T
〈u 2 〉 loc (T) , where 〈u2 〉 loc (T) =
〈u 2 〉(T) − 〈u 2 〉 harm (T). It is a little over-simplifying to call this one view, as the
definitions proposed for 〈u 2 〉 loc (T) are not completely equivalent - however this has
minor importance in the present context. This second view point leads to
where K is a constant.
(
τ(T) = τ 0 exp
K
〈u 2 〉 loc (T)
)
, (7.3.2)
We refer to this viewpoint as the relaxational hypothesis. The relaxational hypothesis
was originally proposed from a phenomenological relation found between viscosity
and 〈u 2 〉 in selenium [Buchenau and Zorn, 1992]. It has later been supported by Ngai
and coworkers, based on qualitative analysis of data-compilations and an extension
of the coupling model [Ngai, 2000, 2004].