Chemistry Reference
In-Depth Information
TI
mVT
Thermodynamic integration in the constant chemical potential, con-
stant volume, and constant temperature ensemble
TPT Thermodynamic perturbation theory
TPT1-MSA First-order thermodynamic perturbation theory combined with the
mean spherical approximation
1
Introduction
Knowledge of the equation of state of polymeric systems (polymer melts, solutions,
and blends) is a central prerequisite for many applications. Theoretical modeling of
the phase behavior of polymeric systems on the basis of statistical thermodynamics
has been a challenging problem for decades [
1 11
]. The initially [
1
,
2
] proposed lattice
model of Flory and Huggins involves many crude approximations, with well-known
shortcomings [
6
,
10
]; however, in complicated cases (e.g., ternary systems such as a
polymer solution in a mixed solvent or a polymer blend in a single solvent) this
approach might still be the method of choice [
12
]. One represents a (flexible) linear
macromolecule by a (self-avoiding) randomwalk on a (typically simple cubic) lattice,
such that each bead of the polymer occupies a lattice site (multiple occupancy of sites
being forbidden, of course, to model excluded volume interactions between the effec-
tive monomers). The chemical bonds between neighboring monomers of the chain
molecule then are just the links between neighboring lattice sites. Solvent molecules
are often simply represented by vacant sites (V) of the lattice. When one deals with
binary blends, two types of monomers (A and B) occur and, apart from the chain
lengths
N
A
,
N
B
of the macromolecules, which are proportional to their molecular
weights, several interaction parameters come into play. Even if we restrict enthalpic
forces to nearest-neighbor interactions, three types of (pairwise) interaction para-
meters
e
AA
; e
AB
and
e
BB
are introduced, which then can be translated into the well-
known Flory Huggins parameters [
6
,
9
]. Of course, the model can also be generalized
to other chain architectures, e.g., block copolymers [
6
,
9
,
11
,
13
]. However, although
the lattice model underlying Flory Huggins theory [
1 8
] and its generalizations (e.g.,
[
14
]) is an extremely simplified description of any polymeric material, the statistical
thermodynamics of this model is rather involved because the analytic treatments
require mean field approximations and further uncontrolled approximations [
6
,
15
,
16
]. The mean field treatment implies that critical exponents characterizing the
singularities of the equation of state near critical points are those [
6
]oftheLandau
theory [
17
]. Studying the Flory Huggins lattice model by Monte Carlo (MC) simula-
tion methods [
18 20
], one avoids these approximations and obtains the correct Ising-
like critical behavior [
6
,
15
,
16
,
21
], which has also been established experimentally
for the critical points of both polymer solutions and polymer blends (see, e.g., [
22
,
23
]). Also, far from the critical region, the approximate counting of nearest-neighbor
contacts between different chains (note that intrachain contacts do not contribute to
phase separation) invalidate simple relations between the basic energy parameters
