II (i ¼ II), i.e., f I ¼ N
I
H = N
I
H þ N
I
D
À
Á
and f II ¼ N
II
H = N
II
H þ N
II
D
À
Á
where N H and N D are
the number of chains of type H or D inside the respective micelles. Considering
proteated and deuterated unimers (number denoted by U H and U D , respectively)
undergoing insertion/expulsion between the micelles, the rate equations can be
written:
dN
I
H
dt
¼ k
H
þ U H N
I
H þ N
I
D
À
Á À k À N
I
H
(101)
dN
I
D
dt
¼ k
D
þ U D N
I
H þ N
I
D
À
Á À k À N
I
D
(102)
dN
II
H
dt
¼ k
H
þ U H N
II
H þ N
II
D
À
Á À k À N
II
H
(103)
dN
II
D
dt
¼ k
H
þ U D N
II
H þ N
II
D
À
Á À k À N
II
H
(104)
From mass-conservation we have:
dN
I
H
dt
þ
N
II
H
dt
¼ 0
(105)
dN
I
D
dt
þ
N
II
D
dt
¼ 0
(106)
It is useful to introduce the relative fraction of H-chains, r:
r ¼
N
I
H þ N
II
H
N I
H þ N II
H þ N I
D þ N II
D
(107)
From Eqs. 101–107:
k
H
þ
k H
À
¼
r
U H
(108)
k
D
þ
k D
À
¼
1 À r
U D
(109)
The time derivative of f I and f II are then obtained by direct derivation and yields:
df I
dt
¼
1
N I
H þ N I
D
Á ð1 À f I Þ
dN
I
H
dt
À f 1
dN
I
D
dt
!
(110)
Kinetics of Block Copolymer Micelles Studied by Small-Angle Scattering Methods
107