structure factors for polymer systems - NIST Center for Neutron ...

0 ⎡ U1
+ u + W11v1n
1(
Q)
+ W12v
2n
2 ( Q)
⎤
v1
n1
( Q)
= −S11(
Q)
⎢
⎥ (38a)
⎣
k BT
⎦
v
2
n
2
( Q)
= −S
0
22
v1 1
2 2
⎡ U 2 + u + W21v1n
1(
Q)
+ W
( Q)
⎢
⎣
k BT
10
22
v
2
n
2
( Q)
⎤
⎥ (38b)
⎦
n ( Q)
+ v n ( Q)
= 0 . (38c)
The last equation represents the incompressibility constraint. The non-interacting or
“bare” structure factors S ( Q)
0
11 and S ( Q)
0
22 have been defined. These equations have
assumed that no copolymers are present in the homogeneous mixture; i.e., that
0
0
S ( Q)
= S ( Q)
= 0 .
12
21
In order to solve the set of linear equations, we extract the perturbing potential u from the
second equation and substitute it into the first equation to obtain:
0 ⎡U
1 − U 2
⎤
1 n1
( Q)
= −S11(
Q)
⎢ + v11(
Q)
n ( Q)
⎥ . (39)
⎣ k BT
⎦
v 1
This applies along with the following equation representing the response of the fully
interacting system:
⎡U 1 − U 2 ⎤
v1
n1
( Q)
= −S11(
Q)
⎢ ⎥ . (40)
⎣ k BT
⎦
The factor v11(Q) and the Flory-Huggins interaction parameter χ12 are defined as:
v
χ
11
12
( Q)
1
2χ
12
= −
(41)
0
S22
( Q)
v 0
W12
⎛ W11
+ W
= − ⎜
k BT
⎝ 2k
BT
22
Here v0 is a reference volume (often taken to be v 0 = v1v
2 ).
⎟ ⎞
.
⎠
The RPA result for a homogeneous binary blend mixture follows:
1 1 1 2χ
S ( Q)
11
12
= + −
(42)
0
0
S11(Q)
S22
(Q) v 0