The Prigogine–Defay ratio and the microscopic theory
of supercooled liquids
R. M. Pick
a͒
IMPMC, Université P. et M. Curie et CNRS-UMR 7590, Paris (F) F-75015, France
͑Received 16 April 2008; accepted 21 July 2008; published online 30 September 2008͒
Prigogine–Defay ratios and, more recently, their frequency extension have been proposed to be a
measure of the number of nonmacroscopic processes involved in the relaxation dynamics of
supercooled liquids. We show that the microscopic theory of the Navier–Stokes equations of those
liquids provides a consistent thermodynamic framework in which all possible dynamical Prigogine–
Defay ratios can be expressed in terms of the same relaxation functions and that these ratios provide
less information than the microscopic theory itself. The latter shows that more than one relaxation
process is certainly always involved in this relaxation dynamics, whatever is the molecular
dynamics, or experimental, technique used to determine the latter. © 2008 American Institute of
Physics. ͓DOI: 10.1063/1.2969899͔
I. INTRODUCTION
Most liquids, when cooled, e.g., at constant pressure,
exhibit a first order transition at a freezing temperature, T
f
,at
which they transform into a crystal. However, some of them
miss this transition and remain, upon further cooling, in a
metastable, called supercooled, state. The most spectacular
manifestation of this state is the constant and dramatic in-
crease in the liquid shear viscosity with decreasing tempera-
ture. The so-called liquid-glass transition takes place at the
arbitrarily defined temperature T
g
at which this viscosity
reaches 10
13
P.
This viscosity variation is due to the corresponding in-
crease, upon cooling, of a time scale loosely defined as the
“structural relaxation time” and this increase is reflected in
many dynamical and/or thermodynamic quantities. In the
early 50s, the question of the number of processes involved
in the relaxation of these supercooled liquids was raised.
Davies and Jones,
1
and later Prigogine and Defay,
2
proposed
an isobaric method to investigate the question. They dis-
cussed the situation in which a supercooled liquid is
quenched so rapidly at a temperature T͑ϵT
m
͒ that it falls out
of equilibrium. It then relaxes at constant temperature and at
the constant pressure P toward another metastable state. In
the spirit of these authors, the out of equilibrium, or glassy,
state must be characterized during the whole relaxation pro-
cess by those two variables plus a set of r, undefined, addi-
tional variables, each of them characterizing a relaxation pro-
cess. Calling ⌬C
p
the difference between the specific heat at
constant pressure per unit volume at equilibrium and its
value at time t=0, i.e., when the liquid was in the glassy
state, calling ⌬
T
and ⌬
␣
P
the similar quantities for the com-
pressibility at constant temperature and for the thermal ex-
pansion coefficient at constant pressure, they defined the so-
called Prigogine–Defay ratio,
⌸
P,T
=
⌬
T
⌬C
P
͓⌬
␣
P
͔
2
T
m
. ͑1.1a͒
They showed that, within this scheme, ⌸
P,T
is equal to unity
only if r=1 ͑one relaxation process, or channel͒, while it is
larger than unity for r Ͼ 1 ͑several relaxation channels͒.In
the present paper, we shall make use of a different, but
equivalent formulation through the quantity
D
P,T
=
m
2
ͫ
⌬͑−
T
͒
⌬͑− C
P
͒
T
m
− ͓⌬
␣
P
͔
2
ͬ
, ͑1.1b͒
where
m
is the mean mass density of the liquid. D
P,T
is
equal to zero if ⌸
P,T
is equal to unity ͑r = 1 case͒ and is
strictly positive for rϾ 1.
References 1 and 2 also extended their technique to the
other sets of conjugate thermodynamic variables that may
appear in the different thermodynamic potentials and they
derived similar results. For instance, in a relaxation experi-
ment performed at constant density and temperature , i.e., for
the variables
and T, where
is the mass density, their result
read, with the formulation of Eq. ͑1.1b͒,
D
,T
= ⌬͑
m
−2
T
−1
͒
⌬͑− C
V
͒
T
m
− ͓⌬

͔
2
ജ 0, ͑1.2͒
where C
V
is the specific heat at constant volume per unit
volume and

is the thermal pressure coefficient; D
,T
is
equal to zero for a single relaxation channel ͑r =1͒ and is
positive otherwise ͑r Ͼ 1͒.
The interest in the Prigogine–Defay ratio, Eq. ͑1.1a͒, has
been recently revived through a paper by Ellegaard et al.
3
These authors noted that, from a phenomenological point of
view, in a supercooled liquid, the usual thermodynamic rela-
tions between the variation of two thermodynamic variables
and the corresponding variation of the conjugate functions
transform into time convolution products. For instance, in
the case of the P and T variables, those relations can be
written as
a͒
Electronic mail: robert.pick@courriel.upmc.fr.
THE JOURNAL OF CHEMICAL PHYSICS 129, 124115 ͑2008͒
0021-9606/2008/129͑12͒/124115/7/$23.00 © 2008 American Institute of Physics129, 124115-1