Chemistry Reference
In-Depth Information
30
25
20
feed
gel
sol
15
10
5
0
0
200
400
600
800
1000 1200 1400 1600 1800 2000
r
Fig. 2 Fractionation effect for homopolymers
initial polymer, but the first droplets of the formed coexisting new phase never do
(with the exception of the critical point) and, hence, they are not located at the
cloud-point curve but on the shadow curve. In the case of statistical copolymers, the
distribution function according to the chemical composition also differs in the two
phases; however, how this distribution changes cannot be predicted a priori. The
residence (sol or gel phase) of the molecules with a high value of
y
depends on the
thermodynamic properties of the selected solvent mixture.
To perform phase equilibrium calculations, the starting point is the segment-
molar chemical potential,
m
i
, related to the segment-molar Gibbs free energy
of mixing. According to the well-known Flory Huggins lattice theory [
54
], the
segment-molar chemical potential (m
i
) for the solvents A and B reads:
1
r
i
1
r
i
1
r
M
m
i
¼
m
i
0
ð
T
;
P
Þþ
RT
ln
X
i
þ
þ
RT
ln
f
i
i
¼
A
;
B
;
(2)
where the first term represents the segment-molar chemical potential of the pure
solvents at system temperature
T
and system pressure
P
. The second term on the
right hand side is the Flory Huggins contribution (with
f
i
¼
1), accounting for the
difference in molecular size. In order to describe the deviation from a Flory
Huggins mixture (with
f
i
¼
1), the segment-molar activity coefficients,
f
i
, are
introduced. The number-average segment number,
r
M
,in(
2
) is for a ternary system,
built up from solvent, nonsolvent, and copolymer, and is given by:
1
ð
1
X
A
r
A
þ
X
B
r
B
þ
X
r
N
¼
X
A
r
A
þ
X
B
r
B
þ
XW
ð
r
;
y
Þ
1
r
M
¼
d
y
d
r
;
(3)
r
0
0
