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e
AA
; e
AB
and
e
BB
and the corresponding Flory Huggins parameters in the expression
describing the free energy of mixing [
6
,
15
,
16
]. If the latter is adjusted to describe the
simulation “data”, a nontrivial dependence of the Flory Huggins parameter(s) on
temperature and volume fractions
ðf
A
; f
B
Þ
of the two types of monomers in a blend
results [
9
,
16
](notethat
f
A
þ f
B
¼
f
V
,where
f
V
is the volume fraction of
vacant sites, which may represent “free volume” [
24
]orsolvent[
6
]).
When one fits the Flory Huggins theory to experiment [
7
], nontrivial depen-
dence of Flory Huggins interaction parameters on temperature and volume frac-
tions also result, but might have other reasons than those noted above: in particular,
it is important to take into account the disparity between size and shape of effective
monomers in a blend, and also the effects of variable chain stiffness and persistence
length [
25
,
26
]). To some extent, such effects can be accounted for by the lattice
cluster theories [
27 30
], but the latter still invokes the mean-field approximations,
with the shortcomings noted above. In the present article, we shall focus on another
aspect that becomes important for the equation of state for polymer materials
containing solvent: pressure is an important control parameter, and for a sufficiently
accurate description of the equation of state it clearly does not suffice to treat the
solvent molecules as vacant sites of a lattice model. In most cases it would be better
to use completely different starting points in terms of off-lattice models.
A basic approach for the description of polymer chains in the continuum is the
Gaussian thread model [
26
,
31
]. Treating interactions among monomers in a mean-
field-like fashion, one obtains the self-consistent field theory (SCFT) [
11
,
32 36
]
which can also be viewed as an extension of the Flory Huggins theory to spatially
inhomogeneous systems (like polymer interfaces in blends, microphase separation in
block copolymer systems [
11
,
13
], polymer brushes [
37
,
38
], etc.). However, with
respect to the description of the equation of state of polymer solutions and blends in
the bulk, it is still on a simple mean-field level, and going beyond mean field to
include fluctuations is very difficult [
11
,
39 42
] and outside the scope of this article.
A powerful theory that combines the Gaussian thread model of polymers with
liquid-state theory is the polymer reference interaction site model [
43
,
44
]. This
approach accounts for the de Gennes [
5
] “correlation hole” effect, and chemical
detail can be incorporated (in the framework of somewhat cumbersome integral
equations that are difficult to solve and need various approximations to be tracta-
ble). Also, in this theory the critical behavior always has mean field character. The
same criticism applies to the various versions of the statistical associating fluid
theory (SAFT) [
45 54
], which rely on thermodynamic perturbation theory (TPT)
with respect to the treatment of attractive interactions between molecules (or the
beads of polymer chains). Some of those theories [
50
,
52
,
53
] seem to perform
rather well when one restricts attention to the region far away from critical
points, as a comparison with the corresponding MC simulation shows [
53
,
55
,
56
]. We shall discuss these comparisons in Sects.
4.1
and
4.2
. Other variants of this
approach, such as the perturbed chain statistical associating fluid theory (PC-SAFT)
approach [
51
]include additional approximations and therefore sometimes suffer
from spurious results, such as artificial multiple critical points in the phase diagram
[
54
]. For this reason, this approach will not be emphasized here.
1
