Chemistry Reference
In-Depth Information
the B polymer segments it encounters. The second term comes from the tendency of
the system to pull polymer into regions where the total density (
r
A
þ r
B
) is less
than
r
0
, and push it out of regions of density greater than
r
0
. The inverse compress-
ibility has been proven to be the proper account for this tendency [
27
,
28
].
To obtain the self-consistent configurations of this system, one should focus on
the quantity
q
A
(r,
t
), which is essentially the ratio of the density at r of the ends of
polymer molecules of type A and length
Ztb
with 0
1, to the end density in the
bulk A. Since the segment at
Zt
is the origin of two independent random walks, one
of length
Ztb
and one of length
Z
(1
<
t
<
t
)
q
A
(r,
t
).
By summing over all segments, or integrating over
t
, the overall segment density of
Aatr is:
t
)
b
, the relative density is
q
A
(r,1
Z
1
r
A
ð
r
Þ¼
q
A
ð
r
;
1
t
Þ
q
A
ð
r
;
t
Þ
dt
(33)
0
The quantity
q
A
(r,
t
) can also be regarded as the ratio of the partition function of a
polymer molecule that starts at r and has
Zt
steps in the effective mean field
W
A
(r),
to that of a polymer in a zero field region, i.e., in the bulk phase. This ratio satisfies a
modified diffusion equation, which, for the dividing surface at
x
¼
0 and the A-rich
phase at
x
>
0, can be written as:
q
A
ð
b
2
6
r
Z
@
1
q
A
ð
r
;
t
Þ
Þ w
r
B
ð
Þ
r
0
þ z
r
r
A
ð
Þ
r
0
þ
r
r
B
ð
Þ
r
0
r
2
q
A
ð
¼
r
;
t
1
r
;
t
Þ
(34)
@
t
By symmetry, the equation for
q
B
(r,
t
) is:
q
B
ð
b
2
6
r
1
Z
@
q
B
ð
r
;
t
Þ
Þ w
r
A
ð
r
Þ
r
A
ð
r
Þ
r
0
þ
r
B
ð
r
Þ
r
0
2
q
B
ð
¼
r
;
t
r
0
þ z
1
r
;
t
Þ
(35)
@
t
with initial conditions:
q
A
ð
r
;
0
Þ¼
q
B
ð
r
;
0
Þ¼
1
q
A
ð1;
t
Þ¼
q
B
ð1;
t
Þ¼
1
(36)
q
A
ð1;
t
Þ¼
q
B
ð1;
t
Þ¼
0
where it is assumed that the asymptotic regions are pure A or B. Thus, (
33
) (
36
) are
the self-consistent set of equations for the density profile calculation.
These equations have been solved asymptotically for effective infinite degree of
polymerization of the chains, and low isothermal compressibility (
w
/
z !
0), to yield
the density profiles:
r
A
ð
;
wÞ
r
0
x
1
2
1
¼
½
þ
tanh 2
x
ð
=
a
I
Þ
(37)
r
B
ð
;
wÞ
r
0
¼
x
1
2
1
½
tanh 2
x
ð
=
a
I
Þ
