with the radial distance r away from the core r > R c and thus the area per chain will
also be a function of r, i.e., s ¼ s(r).
Following the approach by Zhulina et al. [47], the energy density is thus ΦðrÞ ¼
C F Á k B T=sðrÞ
3=2 and the corona free energy per k B T can be calculated from F corona ¼
Ð R m
R c
ΦðrÞ=k B TsðrÞdr ¼ C F Á
Ð R m
R c
sðrÞ
À1=2 dr. C F is a numerical scaling factor of order
unity.
The area per chain is found to have the following approximate form:
sðrÞ
sðrÞ ¼ s 0
r
R c
2
Spheres
sðrÞ ¼ s 0
r
R c
Cylinders
8
<
:
(15)
These give the following results:
F corona ¼ C F
ð R c þD
R c
sðrÞ
À1=2 dx;
sðrÞ ¼ s 0
r
R c
2
Spheres
sðrÞ ¼ s 0
r
R c
Cylinders
8
<
:
(16)
where s(r) is a function that describes the radial dependence of the grafting density,
s (area available per chain), which on the core surface (r ¼ 0) is equal to s 0 ¼ 4πR
2
c =P
for spherical micelles and s 0 ¼ 2πR c  L/P for cylindrical micelles (where L is the
cylinder length). After some calculus, Zhulina et al. showed that the analytical
expressions of F corona can be written as:
F corona ¼
vC F R c
ffi ffi
s
p
ln 1 þ
l A C H N A sl
À2
A
ð Þ
ðvÀ1Þ=2v
vR c
Spheres
2C F R c
ffi ffi
s
p
1 þ
ð1þvÞÁl A C H N A sl
À2
A
ð Þ
ðvÀ1Þ=2v
2vR c
v=ðvþ1Þ
À 1
"
#
Cylinders
8
> > > <
> > > :
(17)
Here C F and C H are numerical prefactors; l B and l A the effective segment lengths
of insoluble and soluble blocks; N B and N A denote the respective number of repeat
units; and v is the excluded volume parameter controlling the conformation of the
chain. In a good solvent, v takes the well-known value 0.588.
F core corresponds to the elastic energy associated with stretching the chains beyond
their unperturbed end-to-end distance to the radius of the core. If this contribution is to
be determined accurately, the fraction of chains needed to stretch must be evaluated.
This was carefully worked out for various geometries by Semenov [39], who also
calculated the fraction of chain-ends needed to effectively fill the core for each
geometry. The results are:
F core ¼ k j Á
R
2
c
R 2
ee
;
k j ¼
π
2
16
Spheres
k j ¼
3π
2
80
Cylinders
(
(18)
Kinetics of Block Copolymer Micelles Studied by Small-Angle Scattering Methods
65
also be a function of r, i.e., s ¼ s(r).
Following the approach by Zhulina et al. [47], the energy density is thus ΦðrÞ ¼
C F Á k B T=sðrÞ
3=2 and the corona free energy per k B T can be calculated from F corona ¼
Ð R m
R c
ΦðrÞ=k B TsðrÞdr ¼ C F Á
Ð R m
R c
sðrÞ
À1=2 dr. C F is a numerical scaling factor of order
unity.
The area per chain is found to have the following approximate form:
sðrÞ
sðrÞ ¼ s 0
r
R c
2
Spheres
sðrÞ ¼ s 0
r
R c
Cylinders
8
<
:
(15)
These give the following results:
F corona ¼ C F
ð R c þD
R c
sðrÞ
À1=2 dx;
sðrÞ ¼ s 0
r
R c
2
Spheres
sðrÞ ¼ s 0
r
R c
Cylinders
8
<
:
(16)
where s(r) is a function that describes the radial dependence of the grafting density,
s (area available per chain), which on the core surface (r ¼ 0) is equal to s 0 ¼ 4πR
2
c =P
for spherical micelles and s 0 ¼ 2πR c  L/P for cylindrical micelles (where L is the
cylinder length). After some calculus, Zhulina et al. showed that the analytical
expressions of F corona can be written as:
F corona ¼
vC F R c
ffi ffi
s
p
ln 1 þ
l A C H N A sl
À2
A
ð Þ
ðvÀ1Þ=2v
vR c
Spheres
2C F R c
ffi ffi
s
p
1 þ
ð1þvÞÁl A C H N A sl
À2
A
ð Þ
ðvÀ1Þ=2v
2vR c
v=ðvþ1Þ
À 1
"
#
Cylinders
8
> > > <
> > > :
(17)
Here C F and C H are numerical prefactors; l B and l A the effective segment lengths
of insoluble and soluble blocks; N B and N A denote the respective number of repeat
units; and v is the excluded volume parameter controlling the conformation of the
chain. In a good solvent, v takes the well-known value 0.588.
F core corresponds to the elastic energy associated with stretching the chains beyond
their unperturbed end-to-end distance to the radius of the core. If this contribution is to
be determined accurately, the fraction of chains needed to stretch must be evaluated.
This was carefully worked out for various geometries by Semenov [39], who also
calculated the fraction of chain-ends needed to effectively fill the core for each
geometry. The results are:
F core ¼ k j Á
R
2
c
R 2
ee
;
k j ¼
π
2
16
Spheres
k j ¼
3π
2
80
Cylinders
(
(18)
Kinetics of Block Copolymer Micelles Studied by Small-Angle Scattering Methods
65