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( )
( ) ∑∑∑
3
2
2 2 2
qq 32π
1/
2 Nρ
Qi
Q j Qk
A 3B
= 2,
002π
xix
j xk
mim
jmk
x pi x pj x pk I
2
3 3 3 3,
ijk
2,
025
kBT
i j k
σ ijσ
ikσ
jk
Modeling
(III.18)
The integrals I are calculated using molecular dynamic results for a pure Lennard-
Jones fluid, and the resulting values were fitted to simple functions of reduced density and
temperature as reported by Gubbins and Two (1978).
As can be observed from Equations III.16 to III.18, when the quadrupole moment is
explicitly taken into account, two more parameters have to be considered in the model, the
quadrupolar moment Q (C.m 2 ) and xp, defined as the fraction of segments in the chain that
contain the quadrupole as will be discussed latter in session III.4.3.
III.2.2. The Crossover Approach
As any other equation of state, soft-SAFT is unable to correctly describe the scaling
of thermodynamic properties as the critical point is approached giving systematically
higher predictions near this point. The mean-field equations of state provide a reasonable
description of fluid equilibrium properties far away from the critical point. However, near
the critical point, due to density and/or concentration fluctuations caused by long-range
correlation between molecules, the thermodynamic properties of a fluid show singularities
and therefore, classical analytical equations fail. Attempts to deal with this problem include
the rescaling of molecular parameters so that the experimental critical point of the pure
compound is matched (Blas and Vega, 1998b; McCabe and Jackson, 1999 and Pàmies and
Vega, 2002). The simple rescaling of molecular parameters improves predictions of the
critical behaviour of several mixtures, but a new set of different molecular parameters for
each family of components is required for the near-critical region. This approach can be
viewed as a practical one but it does not correctly solve the problem since the fluctuations
inherent to the critical region are still ignored in this case.
Renormalization-group methods (RG) have been very successful in describing the
properties of systems near their critical point (Wilson, 1971; Wilson and Fischer, 1972).
However, for engineering applications, the ideal situation would be a model that could
correctly describe, using the same set of parameters and equation, the thermodynamics of
the system far and close to the critical point.
There are different approaches in which the long-wavelength density fluctuations
can be taken into account in the near-critical region, searching for a global equation for real
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