Chemistry Reference
In-Depth Information
chemical composition was neglected. First, Litmanovich and Shtern [
52
] modeled
stepwise copolymer fractionation, where both polydispersities were considered.
Later, Ogawa and Inaba [
53
] also suggested a similar model.
This contribution aims at the development of a theoretical tool for optimization
of copolymer fractionation in columns, where both polydispersities are completely
taken into account.
2 Theory
2.1 Liquid-Liquid Phase Equilibrium of Copolymer Solutions
Fractionations are usually carried out using a solvent (A), a nonsolvent (B) and the
polymer to be fractionated. This means that, for phase equilibrium calculations, a
ternary system must be investigated. Due to the very large number of different
chemical species, the composition of polydisperse systems is not described by the
mole fraction of the individual components, but by a continuous distribution
function. In the case of statistical copolymers, a two-dimensional distribution
function according the molecular mass and the chemical composition must be
used. Usually in polymer thermodynamics, all molecules are imagined to be
divided into segments of equal size. With a standard segment defined as the ratio
of the van der Waals volume of the considered species and the van der Waals
volume of an arbitrary chosen species (for instance one of the solvents or one of the
monomers), a segment number
r
can be defined for each kind of molecule. The
introduction of the segment number leads to segment-molar physical quantities.
The chemical composition of a copolymer, built up from two different monomers,
can be described by the variable
y
. It is given by the ratio of the segment number of
one monomer and the sum of the segment numbers of both monomers, and hence
y
is related to the amount of one monomer in the copolymer. The intensive distribution
function,
W
(
r
,
y
), has to fulfill the normalization condition:
1
ð
1
W
ð
r
;
y
Þ
d
y
d
r
¼
1
:
(1)
0
0
W
(
r
,
y
)d
y
d
r
represents the segment fraction of all copolymer species having seg-
ment numbers between
r
and
r
+d
r
and chemical compositions between
y
and
y
+d
y
.
Due to the polydispersity, the demixing behavior becomes much more compli-
cated for a polydisperse polymer in comparison with a monodisperse polymer, as
shown in Fig.
1
. The binodal curve in this system splits into three kinds of curves: a
cloud-point curve, a shadow curve, and an infinite number of coexistence curves.
The meaning of these curves becomes clear if one considers the cooling process.
