78
T. A. Deaton et al.
formed respectively if p ≤
1
3
,
1
3
< p ≤
1
2
, and
1
2
< p ≤ 1 [60]. Hence, as the value
of the packing parameter increases, BCPs-based aggregates undergo morphological
transition from spherical micelles to cylindrical micelles and further to vesicles.
Numerous theoretical studies examined the micellization of BCPs. Most of these
studies have focused on spherical micelles encompassing amphiphilic AB diblocks.
There are a number of review articles and books that provided an extensive chronological summary of the theoretical research in the field of BCP micellization [60–
64]. One of the key scientific questions that has been investigated is the dependence
of various characteristic parameters of a micellar system, such as the aggregation
number Z (the number of chains constituting one micelle), the radius of gyration
(R g,m ), the core (R g,c ), and the thickness of the corona (H corona ) of the micelle, on
the molecular properties of the BCPs. These molecular properties include the degree
of polymerization of the solvophilic block (N A ) and solvophobic block (N B ). In this
section, a concise overview of the main results of BCP micellization theories will be
discussed. In addition, a brief summary of theoretically-derived power law dependencies between the characteristic parameters, which can be used to compare with
experimental or computational results, will be presented. Some of the power law
scaling functions are derived based on the scaling theories while others are based on
the development of numerical self-consistent mean-field theories or semi-analytical
mean-field models [49, 60, 62]. It is relevant to note that the following relationships are largely focused on the self-assembly of micelles in the dilute to semi-dilute
regime. Limitations can exist for BCP self-assembly scaling relations, specifically
as they translate to relatively large changes concentration [65–67].
2.1 Neutral BCP-Based Micelles
In scaling theories for neutral AB diblock copolymers, the two limiting cases of
spherical micelles are defined as star-like (H corona R g,c ) micelles and crew-cut
(H corona R g,c ) micelles [62]. For star-like spherical micelles, scaling functions for
aggregation number Z and corona thickness H corona have been derived by Daoud and
Cotton [61] as:
Z ∼ N
4/5
B
(2)
H corona ∼ N
3/5
A Z
1/5
(3)
Similar scaling theory results were obtained by Zhulina and Birshtein [68] and
Halperin [69]. The scaling relation between the radius of gyration of the micelle R g,m
and the degree of polymerization N A and N B was predicted by Halperin [69] to be:
R g,m ∼ N
3/5
A N
4/25
B
(4)
T. A. Deaton et al.
formed respectively if p ≤
1
3
,
1
3
< p ≤
1
2
, and
1
2
< p ≤ 1 [60]. Hence, as the value
of the packing parameter increases, BCPs-based aggregates undergo morphological
transition from spherical micelles to cylindrical micelles and further to vesicles.
Numerous theoretical studies examined the micellization of BCPs. Most of these
studies have focused on spherical micelles encompassing amphiphilic AB diblocks.
There are a number of review articles and books that provided an extensive chronological summary of the theoretical research in the field of BCP micellization [60–
64]. One of the key scientific questions that has been investigated is the dependence
of various characteristic parameters of a micellar system, such as the aggregation
number Z (the number of chains constituting one micelle), the radius of gyration
(R g,m ), the core (R g,c ), and the thickness of the corona (H corona ) of the micelle, on
the molecular properties of the BCPs. These molecular properties include the degree
of polymerization of the solvophilic block (N A ) and solvophobic block (N B ). In this
section, a concise overview of the main results of BCP micellization theories will be
discussed. In addition, a brief summary of theoretically-derived power law dependencies between the characteristic parameters, which can be used to compare with
experimental or computational results, will be presented. Some of the power law
scaling functions are derived based on the scaling theories while others are based on
the development of numerical self-consistent mean-field theories or semi-analytical
mean-field models [49, 60, 62]. It is relevant to note that the following relationships are largely focused on the self-assembly of micelles in the dilute to semi-dilute
regime. Limitations can exist for BCP self-assembly scaling relations, specifically
as they translate to relatively large changes concentration [65–67].
2.1 Neutral BCP-Based Micelles
In scaling theories for neutral AB diblock copolymers, the two limiting cases of
spherical micelles are defined as star-like (H corona R g,c ) micelles and crew-cut
(H corona R g,c ) micelles [62]. For star-like spherical micelles, scaling functions for
aggregation number Z and corona thickness H corona have been derived by Daoud and
Cotton [61] as:
Z ∼ N
4/5
B
(2)
H corona ∼ N
3/5
A Z
1/5
(3)
Similar scaling theory results were obtained by Zhulina and Birshtein [68] and
Halperin [69]. The scaling relation between the radius of gyration of the micelle R g,m
and the degree of polymerization N A and N B was predicted by Halperin [69] to be:
R g,m ∼ N
3/5
A N
4/25
B
(4)