The corresponding free energy profile is shown in Fig. 4.
For one-dimensional stationary flow in a potential well, Kramers’ rate theory
gives the following expression for the outgoing flux, J:
J ¼ ÀD exp ÀFðrÞ=k B T
ð
Þ
@
@r
ϕðrÞ exp FðrÞ=k B T
ð
Þ
(30)
where F(y) is the free energy profile along the “reaction coordinate”, y. As a
polymer is expected to follow a curvilinear rather than a straight path, y is not
necessarily the spatial coordinate r as for rod-like surfactants. ϕ(y) is the local
polymer concentration of the diffusing block copolymer characterized by a typical
diffusion coefficient D. Halperin and Alexander demonstrated that the flux could be
rewritten as:
J $ expðÀF
Ã
=k B TÞv diffusion
(31)
where v diffusion is the chain velocity over the barrier and F
*
¼ F(y
* ) is the maximum
free energy. This is principally determined by the interfacial energy penalty due to
the expelled B-blocks, i.e., F
Ã
$ r
2
bud γ $ N
2=3
B l
2
B γ, with r bud being the radius of the
collapsed B-block.
For micelles with a thin corona, N B ( N A , v diffusion is roughly determined by the
time, τ B , necessary to diffuse the length of its insoluble block, i.e., τ B $ N
2=3
B =D.
Assuming classical diffusion of polymer segments in homogeneous surroundings,
Stoke–Einstein’s law gives D $ 1=N
1=3
B l B and τ B $ N
2=3
B l
2
B = N
À1
B l
À1
B
À
Á ¼ N B l
3
B ¼ v B
where v B is the molecular volume of the B-block.
The above-mentioned expression is valid whenever the core is large compared to
the corona. In the opposite limit, as in star-like micelles, the diffusion through the
corona has to be considered instead. Using the Langevin equation to describe the
Free Energy
E a
R c
D
Fig. 4 Illustration of the
chain expulsion process of a
single chain from a star-like
micelle with core radius R c
and corona thickness D and a
corresponding schematic free
energy profile, F(y), along the
reaction coordinate. In the
calculations given in the text
the reference state is chosen
according to F(P + 1) 0 so
that F
* ¼ E a for the expulsion
process
Kinetics of Block Copolymer Micelles Studied by Small-Angle Scattering Methods
73
For one-dimensional stationary flow in a potential well, Kramers’ rate theory
gives the following expression for the outgoing flux, J:
J ¼ ÀD exp ÀFðrÞ=k B T
ð
Þ
@
@r
ϕðrÞ exp FðrÞ=k B T
ð
Þ
(30)
where F(y) is the free energy profile along the “reaction coordinate”, y. As a
polymer is expected to follow a curvilinear rather than a straight path, y is not
necessarily the spatial coordinate r as for rod-like surfactants. ϕ(y) is the local
polymer concentration of the diffusing block copolymer characterized by a typical
diffusion coefficient D. Halperin and Alexander demonstrated that the flux could be
rewritten as:
J $ expðÀF
Ã
=k B TÞv diffusion
(31)
where v diffusion is the chain velocity over the barrier and F
*
¼ F(y
* ) is the maximum
free energy. This is principally determined by the interfacial energy penalty due to
the expelled B-blocks, i.e., F
Ã
$ r
2
bud γ $ N
2=3
B l
2
B γ, with r bud being the radius of the
collapsed B-block.
For micelles with a thin corona, N B ( N A , v diffusion is roughly determined by the
time, τ B , necessary to diffuse the length of its insoluble block, i.e., τ B $ N
2=3
B =D.
Assuming classical diffusion of polymer segments in homogeneous surroundings,
Stoke–Einstein’s law gives D $ 1=N
1=3
B l B and τ B $ N
2=3
B l
2
B = N
À1
B l
À1
B
À
Á ¼ N B l
3
B ¼ v B
where v B is the molecular volume of the B-block.
The above-mentioned expression is valid whenever the core is large compared to
the corona. In the opposite limit, as in star-like micelles, the diffusion through the
corona has to be considered instead. Using the Langevin equation to describe the
Free Energy
E a
R c
D
Fig. 4 Illustration of the
chain expulsion process of a
single chain from a star-like
micelle with core radius R c
and corona thickness D and a
corresponding schematic free
energy profile, F(y), along the
reaction coordinate. In the
calculations given in the text
the reference state is chosen
according to F(P + 1) 0 so
that F
* ¼ E a for the expulsion
process
Kinetics of Block Copolymer Micelles Studied by Small-Angle Scattering Methods
73