Self-Assembled Structures of Amphiphilic Ionic Block Copolymers: Theory, Self-Consistent Field Modeling and Experiment - Self Organized Nanostructures of Amphiphilic Block Copolymers - page 88

7.1 Structural Transitions in Starlike Micelles with Annealing

PE Corona

The block copolymer chains with a strongly asymmetric composition, N A
γ

7 v − 3 / A , form starlike micelles at an arbitrary small degree of ion-
ization of the coronal block. The free energy per chain in a starlike micelle with an
annealing PE coronal block is given by:

2 / 7 5 /

( N B / ϕ )

3 N B
ϕ

4 / 9

N A v A c p +

1

2

1

F 9

2 γ k B T =

N − 1 / 9
A

c − 2 / 9

2 /

+ − +( α c p / Φ ion )

p

+ ln ( 1 − α ) , (100)

where the degree of ionization is determined from ( 97 ).

Further analysis of the free energy as a function of c p (or

α ( α b , Φ , c p )

ion

α ), ( 100 ) and( 97 ),

indicates the existence of a single minimum at both low, Φ α b c p , and high,

ion

Φ α b c p , salt concentrations. This minimum corresponds to a single population

of micelles that coexist with unimers either at low or at high salt concentration in
the solution. At intermediate salt concentrations, F

ion

might exhibit two minima
as a function of c p . The presence of two minima in the free energy indicates the
possibility of abrupt (quasi-first order) salt- and pH-induced structural transitions in
block copolymer micelles. These transitions are discussed below in Sect. 7.1.3 .

( c p )

7.1.1 Quasi-neutral Micelles

At low salt concentrations, Φ ion α b c p , the ionization of A segments in the micelle

corona is low, α α b . In this case, the excluded-volume repulsions (binary inter-

actions) give the dominant contribution to the free energy of the corona. As a result,
the structural properties of the quasi-neutral micelles approximately coincide with
those formed by non-ionic block copolymers and are given by ( 77 )and( 78 ).
An increase in the salt concentration leads to the progressive replacement of H +
ions in the micellar corona by, e.g., Na + ions of the added salt (i.e., increases the
local pH in the corona). This promotes the increase in the degree of ionization α of

the coronal blocks as:

α b
1 − α b · Φ ion 1 / 2

γ − 3 / 11 N B
ϕ

− 2 / 11

α = N 5 / 11

A

v 9 / 22
A

. (101)

The latter leads to a slight decrease (on the level of the correction terms) in the
aggregation number, and an increase in the radius of the corona.
The free energy per chain in a quasi-neutral micelle is given by:

k B T = γ

11 N 1 / 11
A

v 2 / 11
A

6 / 11 4 /

F quasi − neutral / ( N B / ϕ ) + N A ln ( 1 − α ) . (102)

Here, the first term is dominant because α , given by ( 101 ), is small.