Soft Materials, 3(2&3): 89–120, (2006)
Copyright # Taylor & Francis Group, LLC
ISSN 1539-445X print/1539-4468 online
DOI: 10.1080/15394450600801228

EFFECTS OF ADDED ELECTROLYTES ON THE STRUCTURE OF
CHARGED POLYMERIC MICELLES

S. Mochizuki, S. Komura, and T. Kato
Tokyo Metropolitan University, Tokyo, JapanA

Department of Chemistry,A
We investigate the effects of added electrolytes on the structure of spherical aggregates made ofneutral/charged diblock copolymers with weakly acidic groups on the charged block. Using a simple
model based on the theory of annealed polyelectrolyte brushes, we study the added-electrolyte effects
that differ according to the magnitude of the ionic strength and the acid concentration. They arepresented by the regime diagrams as a function of the ionic strength and the acid concentration.
Depending on the neutral/charged relative block length, six different types of regime diagrams
are obtained. We also discuss the poor solvent case where the backbone of the charged block ishighly hydrophobic. As a result of the competition between the osmotic repulsion and the hydrophobic attraction, the addition of salt induces a reentrant change of the structural parameters.

Keywords Macromolecular and polymer solutions, Polymer melts, Swelling, Supramolecular and biochemical assembly, Polyelectrolytes

INTRODUCTION
Charged diblock copolymers composed of a neutral block and acharged block self-assemble to form micelles in solvents such as water orwater– methanol mixtures that are selective for charged blocks. In a
dilute aqueous solution without added salt, micelles are often spherical,
consisting of the core made up of neutral blocks and the corona composedof extended ionic blocks. Their structural properties have been investigated both experimentally and theoretically in order to understand theproperties of polyelectrolyte brushes and the mechanism of colloidal stabilization by polyelectrolytes (1, 2).
In the previous theoretical approaches, charged blocks in the corona
region are described as polyelectrolyte brushes (i.e., monolayers composedof polyelectrolytes that are grafted at one end onto an impermeablesurface) (3–11). Polyelectrolyte brushes are classified into two types: quenched
Received April 5, 2006; Accepted May 8, 2006.
Address correspondence to S. Komura, Department of Chemistry, Faculty of Science,
Tokyo Metropolitan University, Tokyo 192-0397, Japan. E-mail: komura@comp.metro-u.ac.jp

89

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S. Mochizuki et al.

brushes and annealed brushes (12). For quenched brushes, such as polystyrene
sulfonate brushes, the number and position of charges on the grafted chain
are fixed. For annealed brushes, on the other hand, they are controlled by
the external conditions, such as pH. These brushes are made of polyacids(or polybases) with weakly dissociating groups such as carboxyl groups (–COO2Hþ). The fact that the ionization degree depends on the external conditions leads to nontrivial behaviors of annealed brushes. For example, thethickness of annealed brushes increases with the ionic strength at very low
ionic strength rather than shrinking due to the screening effect (13–16).
For charged micellar solutions, the addition of electrolytes such as salt oracid changes the size of micelles (17–22) or induces morphological changes(22–26). Such phenomena occur due to the weakening of the electrostaticinteractions between the corona chains. In dilute aqueous solutions of polystyrene-b-poly(acrylic acid) (PS-PAA) copolymers, for example, the additionof micromolar HCl or millimolar NaCl can cause the morphology change of
PS-PAA micelles (22). On increasing the electrolyte concentration, the morphology changes from spheres to rods, vesicles, and then to large compoundvesicles. The effects of added electrolytes depend on whether the charged
blocks are quenched or annealed. For the quenched case, the effects ofadded salt are similar to those of added acid, because both of them localize
near the charges on the corona chains (screening effect). For the annealedcase, the addition of salt causes the promotion of dissociation at very low
ionic strength, or the shielding of the ionized corona chains at relatively
high ionic strength (13–16). Added acid, on the other hand, reduces theionization degree of the corona chains by protonating the ionic sites.
Previously, the effects of added salt or acid on charged micellar solutions
were investigated theoretically (5, 8–11). It was predicted that, for quenchedcopolymers, the aggregation number is constant at very low salt concenabove a threshold salt concentration (9).
tration, but it increases as C6/11
SThis behavior is in qualitative agreement with the result of the scatteringexperiment on the aqueous solutions of polyethylethylene-b-poly(styrenesulfonic acid) (20). For annealed copolymers, the coupling between the ionization and the aggregation of copolymers was found theoretically. At very
low ionic strength, for instance, the aggregation number decreases withincreasing the salt concentration due to the promotion of dissociation(10, 11). Such a feature is not exhibited by quenched copolymers.
The purpose of the present paper is to investigate the effects of added
electrolytes on the structure of charged micelles. We consider aggregatesof diblock copolymers composed of a neutral block and a charged blockwith weakly acidic groups (i.e., annealed block). The micelles are assumed
to be spherical, as observed in many experiments. The dependences of
their structural parameters on the salt and acid concentrations are investigated using a simple model based on the theory of annealed polyelectrolyte
brushes. Generally speaking, the interactions between the charged

Effects of Added Electrolytes

91

monomers in a polar solvent include not only the electrostatic repulsion but
also the hydrophobic interaction, because the backbone of the charged blockis made of hydrophobic hydrocarbons. Nevertheless, when the hydrophobicity of the blocks is not too large, the hydrophobic attraction can be neglected with respect to the electrostatic contribution. For such a case (usolvent case), we perform a scaling analysis to investigate the effects ofadded electrolytes on micellar structure. We find that the effects of electrolytes depend not only on the magnitude of the ionic strength and the acidconcentration, but also on the neutral/charged relative block length. On
the other hand, when the charged block is highly hydrophobic (e.g., poly(methacrylic acid) block with a-methyl groups), we need to deal with thepoor solvent case. As a result of the competition between the electrostatic
repulsion and the hydrophobic attraction, we predict a reentrant changeof the micellar structure caused by the addition of salt.
This paper is organized as follows. In the next section, we introduce amodel for a spherical micelle. In the third section, critical micelle concentration is derived by minimizing the free energy of a micellar solution. In
the fourth and fifth sections, we perform a scaling analysis for the structural parameters of crew-cut micelles and star-like micelles, respectively.
In the sixth section, various regimes characterized by different scaling
expressions are organized into six types of regime diagrams. In theseventh section, we numerically calculate the dependences of the structuralparameters on the electrolyte concentration for the poor solvent case.
Conclusions and discussion are provided in the final section.
FREE ENERGY OF A SPHERICAL MICELLE
Consider a charged diblock copolymer composed of a neutral A blockand a charged B block in a selective solvent that is incompatible with the A
block. The polymerization degrees of the A and B blocks are denoted by NA
and NB, respectively. We assume that each repeating unit of the B blockpossesses a weakly acidic group such as a carboxyl group ( – COO2Hþ).
The dissociation degree of the acidic groups a depends on the local
concentration of protons cH:

a
Ka
¼
;
1  a cH

ð1Þ

where Ka is the dissociation constant.
When the concentration of the copolymers exceeds a threshold value
called the critical micelle concentration (CMC), molecules self-assemble
to form micelles. CMC is typically so small that the intermicellar interaction
can be neglected. Micelles are considered to be spherical as observed inmany experiments (2, 18 – 23). As shown in Fig. 1, they consist of a core

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S. Mochizuki et al.

FIGURE 1 A spherical micelle. It consists of a core region (radius Rc) made up of neutral blocks and asurrounding corona region (thickness L) where charged blocks are stretched. The concentration ofmobile ions of species i inside the corona region and in the solution are denoted by ci and Ci, respectively.

region made up of the neutral A blocks and a surrounding corona regionin which the charged B blocks are stretched. We assume that the interface
between these two regions is sharp and that the core region is incompressible. The latter condition is expressed as
4pR3c ¼ pNA a3 ;ð2Þ3
where Rc is the core radius, p the aggregation number, and a the size ofthe repeating unit of the copolymer. In the corona region, the concentration of the B monomers is given by
NB
NB
¼ 3
cp ¼
VB a NA

"

L
1þ
Rc

#1

3
1

;

ð3Þ

where VB is the volume of the corona per chain, and L is the corona thickness. Hence, L is expressed as a function of cp by"
1=3 #NB
L ¼ Rc 1 þ 3
1 :ð4Þ
a NA c p
The total free energy of a spherical micelle per chain is written as asum of three contributions (5, 6, 8– 11, 27, 28):
f ¼ fcore þ finterface þ fcorona :ð5ÞThe core energy fcore represents the elastic energy per A block:fcore ¼3R2c:
2NA a2ð6Þ

Effects of Added Electrolytes

93

Here and in the following, all energies are expressed in units of thethermal energy kBT. In the above, we have assumed that each A block is
uniformly stretched so that its end-to-end distance is equal to the coreradius Rc. The second term finterface represents the interfacial energy
between the core and the corona regions:finterface ¼ g

4pR2c 3gNA a3
¼
;
p
Rc

ð7Þ

where kBTg is the interfacial tension. The value of g a2 is typically of orderunity. We neglect the interfacial energy between the corona and the outersolution, because it is much smaller than the core– corona surface energy.
The form of the corona energy fcorona will be discussed in the following.
We assume that the corona chains are grafted on the core– corona
interface to form an annealed polyelectrolyte brush (10– 14). We thenadapt the model of annealed polyelectrolyte brushes given in Ref. (13).
All the corona chains are assumed to be uniformly stretched up to thecorona thickness L. The corona region is surrounded by the solution ofacid and salt. We require that the electroneutrality condition is satisfied
both in the corona region and in the outer solution. The concentrationof each component is taken to be uniform.
Let ci and Ci be the concentration of mobile ions of species i in thecorona region and the outer solution, respectively. For simplicity, we
assume that the weakly acidic group, acid, and salt are all monovalent.
The electroneutrality conditions in the corona region and the outersolution are written as
X
X
acp þ
ci ¼
c jþ ;
ð8Þ
i

j

X

X

i

Ci ¼

C jþ ;

ð9Þ

jrespectively, where a is the dissociation degree defined in Eq. (1). The
mobile ions are distributed according to the Boltzmann’s law:eFci ¼ Ci exp
;
ð10Þ
kB T


eF
c jþ ¼ C jþ exp 
;
ð11Þ
kB T
where e is the elementary charge, and F is the average electrostatic potential in the corona region with respect to the outer solution. Notice thatF , 0 owing to the negative charges on the corona chains. Hence, theproton concentration in the corona region is higher than that in the

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S. Mochizuki et al.

outer solution (i.e., cH . CH) [see Eq. (11)]. We then obtain from Eq. (1)
Ka
:
ð12Þ
CH
Thus, the dissociation degree of the corona chains is smaller than that of
the isolated charged blocks in the outer solution. Eqs. (10) and (11) giverise to the so-called Donnan rule (29, 30):

a,

C jþ
ci
¼
:
c jþ
Ci

ð13Þ

Using Eqs. (1), (8), (9), and (13), we obtain a relation that determines a:acp
Ka 1  a CH a

¼
:
ð14Þ
CH a
Ka 1  aCi
P
Here, Ci ¼ j Cjþ is the ionic strength in the outer solution. From thisequation, the asymptotic expressions for a can be obtained for the two limiting cases:8Ka Ci 1=2>>
ðCi  acp Þ;
<
CH cp
a
ð15Þ
>
K
>
a
:
ðCi  acp Þ:CHHere, a is taken to be much smaller than unity, because we consider thecase in which the corona chains have weakly dissociating groups underacidic condition. In the low ionic strength limit (Ci  acp), added salt promotes the dissociation of weakly acidic groups. In the high ionic strength
limit (Ci  acp), on the other hand, a is determined by the proton concentration in the outer solution. In the latter case, we have cH  CH [see Eqs.
(1) and (15)].
Under the electroneutrality condition, the bare electrostatic repulsion
between the corona chains is screened by the counterions. The corona freeenergy per chain can be written as a sum of the three contributions (31):
3L2
1 3 2 1 6 3
þ VB  ta cp þ wa cp
fcorona ¼
2
62NB a2
9
20
18
0
1
<
=
X
X
X
X
ci þ
c jþ A log a3 @ci þc jþ A  1
þ VB 4 @
:
;
i
j
i
j
0
@

X
i

Ci  þ

X
j

93
18
0
1
<
=
X
X
C jþ A log a3 @Ci  þC jþ A  1 5:
:
;
i
j
ð16Þ

Effects of Added Electrolytes

95

The first contribution is the elastic energy of the corona chain. The second
contribution represents the nonelectrostatic interactions between the
charged monomers. Notice that ta3 and wa6 denote the second and thethird virial coefficients, respectively. The last contribution is the entropy
loss of the counterions confined within the corona. Using Eqs. (3), (4),(8), (9), and (13), fcorona is rewritten as(1=3 )2
fcorona
3R2c
NB
1
1
¼
1þ 3
1  ta3 cp þ wa6 c2p
2 2
2
6
NB
a
N
c
2NB a
A p





2CH Ci2CH3
log a acp þþ aþ
Ci  1
cH cp
cH


2Ci
flogð2Ci a3 Þ  1g:
cp

ð17Þ

Collecting all the expressions for fcore, finterface, and fcorona, the total
free energy per chain is summarized as23(1=3 )2
 c ; c p Þ
2
f ðR
3R
N
N
A
B
c 4
¼
1þ
1þ
1 5
2NA NBNB
NA cp
NB
3g NA 1
1
 tcp þ wc2p
c 2
6
NB R






2C H C i
2C H Ci  1þ aþ
log ac p þcH cp
c H
þ



2C i
flogð2C i Þ  1g:
cp

ð18Þ

In the above, we have used the following dimensionless quantities:
 c ¼ Rc a1 ;Rg ¼ ga2 ;ci ¼ ci a3 ði ¼ p; HÞ;C j ¼ Cj a3 ð j ¼ H; iÞ:

ð19Þ

Minimizing f with respect to R̄c, we obtain2

 c ¼ N 2=3 g 1=3 41 þ NA
R
A
NB

(

NB
1þ
NA c p

1=3

)2 31=3
1 5
:

ð20Þ

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S. Mochizuki et al.

Hence, the minimized free energy as a function of c̄p only is given by23(1=3 )2 1=3
f ðcp Þ 9 1=3 1 2=3
N
N
1
1
A
B
¼ NA NB g 41 þ1þ
1 5  tc p þ wc2p226NBNA c p
NB


 


2C H C i
2C H 2C i
log acp þCi  1 þ aþflogð2C i Þ  1g:cH cp
c H
cp
ð21Þ
In the limiting cases, the last two terms of Eq. (21) (the entropy terms) canbe approximated as8
1
1  1=2 1=2
> að1 þ log f1c p
ðC i  acp Þ;
ci Þ ¼ ð1 þ log fci Þðb Ci Þfe ðcp Þ <

cp
>
NB
ðC i  acp Þ;
: 2
4b Ci
ð22Þ
respectively, where we have used the following definitions:

fci ¼ acp ; b ¼ CH =Ka :ð23Þ

The approximated expressions for fe indicate that the mobile ions in thecorona induce an effective repulsion between the charged monomers.
This interaction competes with the elastic energy and the nonelectrostatic
attraction between the monomers.
CRITICAL MICELLE CONCENTRATION
When the copolymer concentration exceeds the critical micelle concentration (CMC), the copolymers self-assemble to form micelles. In thissection, we derive the expression for CMC by introducing the freeenergy of a micellar solution. In addition, we obtain the free energy ofan individual copolymer in the solution.
As in the previous section, we assume that the electroneutrality
condition in the corona is satisfied for each micelle. In this case, the bareelectrostatic interaction between the micelles can be neglected. The total
free energy of a dilute micellar solution is written as [6]X
X f p  f p 
fp fp þ
log
1 ;
ð24Þ
Fmic ¼ppppwhere p is the aggregation number as before, fp the volume fraction of
copolymers packed into micelles composed of p copolymers, and fp themicellar energy per copolymer of aggregation number p. When theEffects of Added Electrolytes

97

formation of spherical micelles is favored, fp takes the minimum value
for a certain aggregation number p . The first term of Eq. (24) represents
the average energy of the micelle. The second term denotes the translational entropy of the micelles. The nonelectrostatic interactions between
micelles are neglected.
To determine the optimum size distribution of charged micelles, we
minimize Fmic with respect to ffpg under the constraint of mass conservation:X

fp ¼ f;ð25Þ

pwhere f is the total volume fraction of copolymers in the solution. Using a
Lagrange multiplier method, we obtain the size distribution of micelles as

fp ¼ p½f1 expð f1  f p Þp :ð26Þ

From this equation, the total volume fraction of copolymers is approximately written asf  f1 þ f p ¼ f1 þ p ½f1 expð f1  f p Þ p

ð27Þ

Because fp , 1, we have f1 , exp( fp 2 f1). When f , exp( fp 2 f1), fp isnegligible compared to f1, whereas fp increases rapidly when f .
exp( fp 2 f1). Hence, CMC is given by

fcmc ¼ expð f p  f1 Þ:

ð28Þ

The condition required to form micelles is fcmc , 1.
We next derive the optimum free energy of an individual charged
copolymer f1. In the solution, the neutral A block collapses, while the conformation of the charged B block is a spherical coil of radius L̄1 a (3, 4).
Then the free energy of an unaggregated copolymer is written as8
1=2 3=2
1>< ðNB b C i Þ L 1
þf1 ðL 1 Þ ¼ NA2=3 g þ
NB2
3
2NB >
:
L 1
2
4NA b Ci
2
3L 1ðCi  ac0p Þ;ðCi  ac0p Þ;

ð29Þ

where cp0 is the monomer concentration in the coil of the B block.
The first term represents the interfacial energy of the collapsed A block.
The second term is the elastic energy of the B block. The last term is thetranslational entropy of the mobile ions inside the coil of the B block.
This term is obtained by recalling the discussion in the second section of

98

S. Mochizuki et al.

this paper. Minimizing f1 with respect to L̄1, we obtain(
NB3 b1 C i
ðC i  NB2 b1=2 Þ;L 1 
1=5
NB3=5 b2=5 C i
ðC i  NB2 b1=2 Þ:ð30Þ

Hence, the optimum free energy of a single copolymer is(2ðC i  NB2 b1=2 Þ;NB5 b2 C i
2=3
F1 ¼ NA g þ
2=5ðC i  N 2 b1=2 Þ:N 1=5 b4=5 C
B

i

ð31Þ

B

In the following arguments, the single copolymer states characterized by
1/2
1/2
and C̄i  N22
are referred to as “M0” and “M1,”C̄i  N22
B b
B b
respectively.
CREW-CUT MICELLES
In this and the next two sections, we discuss the situation in which
the non-electrostatic interaction between the charged monomers canbe neglected with respect to the ion-induced repulsion. Hence, we set
t ¼ w ¼ 0. In this case, the free energy of a spherical micelle is obtained
from Eqs. (21) and (22) as
1=3
9 1=3 2=3
NA u21þþ fe ðuÞ;ð32Þ
f ðuÞ ¼ NA g
2
NB
where
8
3
1=2
1
1
>
< ð1 þ log fci Þ½NA NB b C i fð1 þ uÞ  1g
fe ðuÞ ¼
NB2>
fð1 þ uÞ3  1g1:2
4NA b CiðC i  acp Þ;ðC i  acp Þ:ð33ÞHere we have changed the variable from c̄p to u, which is defined byL
NB 1=3
u;
¼ 1þ
1:
ð34Þ
Rc
NA cpAccordingly, the core radius is rewritten as

1=3
NA u2
2=3 1=3
1þ
:
Rc ¼ NA g
NB

ð35Þ

In the following, we discuss two different types of micelles, namely, thecrew-cut micelles (Rc  L or u  1) and the star-like micelles (Rc  L oru  1). By minimizing f(u), we find out various regimes with differentEffects of Added Electrolytes

99

scaling expressions for the structural parameters. These regimes arecharacterized by the magnitude of the ionic strength C̄i and the acid concentration b. Notice that the dissociation degree of the charged block in
the outer solution is very small (Ka/CH ¼ b21  1) because we do not
add any base.
In the following, we discuss the structural parameters of the crew-cut
micelles (u  1). We find out five different regimes (C0, C1, C2, C3, andQN) characterized by the different scaling behaviors.
Low Ionic Strength Limit
Using Eqs. (32) and (33), the free energy of a crew-cut micelle in the
low ionic strength limit (C̄i  ac̄p) is written as
1=3
9 1=3 2=3
NA u21þð1 þ log f1
f ðuÞ ¼ NA g
ci Þ
2
NB
 ð3NA NB b1 C i Þ1=2 u1=2 :ð36Þ

The dissociation degree increases with decreasing the acid concentrationin the outer solution [see Eq. (15)], and so does the osmotic pressure
exerted on the corona chain. Hence, the corona chain is extended when
the acid concentration is low, and the elastic energy of the corona chain
exceeds that of the core chain. Such a situation is characterized by thecondition NAu2/NB  1. In this case (C0 regime), the free energy of acrew-cut micelle is approximated as
9
1  1=2 1=2
u :f ðuÞ ¼ NA2=3 NB1=3 g 2=3 u2=3  ð1 þ log f1
ci Þð3NA NB b Ci Þ2

ð37Þ

Minimization with respect to u leads to3u  NA1 NB5 g 4 b3 C i ;ð38Þ

where we have assumed that the logarithmic term is approximately
constant. Using Eqs. (15), (35), and u ¼ L̄/R̄c, we get c  NA N 3 g 3 b2 C 2 ;
R
B
i

ð39Þ

L  NB2 g 1 b1 C i ;

ð40Þ

2

a  NB2 g 2 b2 C i :ð41Þ

The above results as well as the others are summarized in Table 1. From
Eqs. (40) and (41), we find L̄  NBa1/2. This is the scaling expression for theosmotic brush with thickness determined by the balance between the elastic
force and the osmotic pressure (32). The corona thickness increases with

100

S. Mochizuki et al.

TABLE 1 Scaling expressions for the structural parameters for a charged spherical micelle and a
unaggregated copolymer.
RegimeaR̄c

L̄(L̄1b)

C0
C1
C2
C3
S1
S2
S3
QN

3 2 22
NAN23
B ḡ b C̄i
23/5 3/5 4/5 2/5
NANB ḡ b C̄i
1/3
N2/3
A ḡ
2/3 1/3
NA ḡ
N7/11
N21/11ḡ5/11b4/11 C̄2/11
ABi
2/3 1/3NA ḡ
2/3 1/3
NA ḡ
1/3
N2/3
A ḡ

N2Bḡ21b21 C̄i
NB4/5ḡ1/5b22/5 C̄21/5
i
21/9 21/3 1/3
N1/9
b
C̄i
A NBḡ
N21/9
NBḡ1/9b22/3 C̄21/3
Ai
NA2/11NB6/11ḡ3/11b22/11 C̄21/11
i
3 21 21N21b C̄i
A NBḡ
1/5 3/5 1/5 22/5 21/5
NA NB ḡ b
C̄i
N1/2
B

M0
M1

N1/3
A
N1/3
A

N3Bb21 C̄i
22/5 21/5
N3/5
C̄i
B b

a
N2Bḡ22b22 C̄2i21b22/9 22/3 2/3
N2/9
b
C̄i
A ḡ
21
b
b21
4 22 22 2
N22
b C̄i
A NBḡ
b21
b21 c
22/9 22/3 2/3 c
N2/9
b
C̄i
A ḡ
22 4 22 22 2 c
NA NBḡ b C̄i
N4Bb22 C̄2ib21

aC, S, QN, and M stand for crew-cut, star-like, quasi-neutral micelles and a single copolymer, respectively.
The radius of a spherical coil composed of a charged block.
ca varies according to the magnitude of C̄i and the neutral/charged relative block length (see Sec.
REGIME DIAGRAMS).
b

the ionic strength, because the dissociation degree increases. This behavior ispeculiar to the corona composed of charged blocks with weakly dissociatinggroups (13). For the corona with strongly dissociating groups, its thickness
decreases with increasing the ionic strength due to the screening effect(5, 8, 9, 19, 20). In the C0 regime, u decreases by the addition of acid.
24/3
N3/2
C̄i, the elastic energy ofWhen u  (NB/NA)1/2 or b  N21/6
A
B ḡ
the core becomes dominant. In this case (C2 regime), the free energy of acrew-cut micelle is approximated as
93
f ðuÞ ¼ NA1=3 g 2=3 þ NA4=3 NB1 g 2=3 u2  ð1 þ log f1
ci Þ22
 ð3NA NB b1 C i Þ1=2 u1=2 :ð42Þ

Minimizing f with respect to u, we obtain
1=3

u  NA5=9 NB g 4=9 b1=3 C i :ð43ÞUsing Eqs. (15) and (35), we get c  N 2=3 g 1=3 ;
R
A

ð44Þ

1=3
L  NA1=9 NB g 1=9 b1=3 C i ;

ð45Þ

2=3

a  NA2=9 g 2=9 b2=3 C i ;ð46Þ(see Table 1). As in the C0 regime, this scaling expression for the corona
thickness is identical to that for the osmotic brush (32). Owing to theEffects of Added Electrolytes

101

dominance of the core elastic energy, the core radius depends neither on
the acid concentration nor on the ionic strength. However, the corona
3/2 21/3
C̄i, theshrinks by the addition of acid. At b  b2N ; N1/3
A NB ḡ
1/2
corona thickness takes the minimum size L̄  NB , which is the radius ofa neutral polymer in a u-solvent. For b . b2N, the structural parametersof a crew-cut micelle are constant (quasi-neutral [QN] regime).
As mentioned above, the C0 regime is characterized by three
conditions: the low ionic strength limit (C̄i  ac̄p), the dominance of thecorona elastic energy (NAu2/NB  1), and the crew-cut condition(u  1). Using the results of Eqs. (38) and (41), these conditions can be
rewritten as 1=2 ;C i  NB1 gb

ð47Þ

C i  NA1=6 NB3=2 g 4=3 b;

ð48Þ

C i  NA1=3 NB5=3 g 4=3 b:

ð49Þ

So that these conditions are simultaneously satisfied for b  1, NB should
2/3
1/2
 NB  NA. In the case of N1/2
 NB  NA,
be restricted to N1/3
A ḡ
A ḡ
the transition between the crew-cut and the star-like micelles occurs when
25/3 4/3ḡ b).
u  1 (or, equivalently, C̄i  N1/3
A NB
The four conditions characterizing the C2 regime are obtained byrepeating the same discussion. They are summarized in Table 2. Inorder to fulfill these conditions simultaneously for b  1, NB should be
2/3
2/3. In the case of NA  NB  N4/3
, the tranrestricted to NB  N4/3
A ḡ
A ḡ
sition between the crew-cut and the star-like micelles occurs when u  1
23 4/3b).
(or, equivalently, C̄i  N5/3
A NB ḡHigh Ionic Strength Limit
In the limit of high ionic strength (C̄i  ac̄p), the free energy of a crewcut micelle is written as

1=3
9
NA u2
NB2
f ðuÞ ¼ NA1=3 g 2=3 1 þ
þ
u1 :
ð50Þ
2
NB
12NA b2 C i
At low acid concentration, the elastic energy of the corona is larger thanthat of the core (NA u2/NB  1). In this case (C1 regime), the minimizationof f(u) gives c  NA N 3=5 g 3=5 b4=5 C 2=5 ;
R
i
B

ð51Þ

1=5
L  NB4=5 g 1=5 b2=5 C i :ð52Þ

The dissociation degree is determined by the acid concentration, a ¼ b21[see Eq. (15)]. From these results, we find L̄  NBv1/3s̄21/3 where

102

S. Mochizuki et al.

TABLE 2 Conditions for each regime.
RegimeCondition
(i)a
(ii)b
(iii)c
(iv)d

(i)
(ii)
(iii)
(iv)

(i)
(ii)
(iii)
(iv)

C0

C1

C2

22/3 22C̄i  N25/3
N7/3
b
A
B ḡ
1/2
C̄i  N21
B ḡ b22/3 22C̄i  N25/6
N3/2
b
A
B ḡ

23 4/3C̄i  N5/3b
A NB ḡC̄i  N21/3
ḡ1/3b21/2
A
23/2 4/3
C̄i  N1/6
ḡ b
A NB
C̄i  N21/3
N23/2
ḡ1/3b
A
B

C3

S1

S2

N3Bḡ22/3b22
C̄i  N27/3
AC̄i  N21/3
ḡ1/3b21/2
A
25/6 3/2 22/3 22C̄i  NA NB ḡ
b
1/3 22C̄i  N21/3
N3/2
b
A
B ḡ

22/3 22C̄i  N25/3
N7/3
b
A
B ḡ
3 7/2
C̄i  N2AN25ḡbB
1/2 22/3 22C̄i  N1/6b
A NB ḡ

S3

M0

M1

1/2
C̄i  N22B b

1/2
C̄i  N22
B b

2/3
QN (NB  N4/3
)
A ḡ

2/3
QN (NB  N4/3
)
A ḡ

N23/2
ḡ1/3bC̄i  N21/3
AB
3/2 1/3 22C̄i  N21/3
N
ḡ
b
A
B

C̄i  NAN25/2
ḡ b
B
22
C̄i  NAN1/2
B ḡ b25/3 4/3
C̄i  N1/3
ḡ b
A NB
1/2
C̄i  N21
B ḡb
23/2 4/3
C̄i  N1/6
ḡ b
A NB

N3Bḡ22/3b22
C̄i  N27/3
A
1/2
C̄i  NAN22
B ḡ b
1/2 22/3 22C̄i  N1/6b
A NB ḡ22
C̄i  NAN1/2
B ḡ b23 4/3C̄i  N5/3b
A NB ḡ
1/2
C̄i  NAN22
B ḡ b3 7/2 eC̄i  N2AN25
B ḡ bC̄i  NAN25/2
ḡ b
B

a

Crew-cut micelles (u  1) or star-like micelles (u  1).
Low ionic strength (C̄i  ac̄p) or high ionic strength (C̄i  ac̄p).
c
The dominance of the elastic energy of the core (NAu2/NB  1) or the corona (NAu2/NB  1).
d
The lower limit of the corona thickness (L̄  N1/2
B ).
eThis relation is derived from the condition that the micellar free energy takes the minimum value as
discussed in Appendix A.
bv ¼ a2/C̄i, and s̄ ¼ 3NA/R̄c is the surface area per grafted polymer [see Eq.
(7)]. This is the scaling expression for the thickness of a planar neutralpolymer brush with an effective excluded volume parameter v ¼ a2/C̄i(33, 34). The effect of added electrolytes (salt or acid) is reflected in the
excluded volume parameter, as in the case of the corona with strongly dissociating groups (8). By increasing the acid concentration, the corona
shrinks because the effective excluded volume decreases. The corechains, on the other hand, are more stretched by adding acid. Eventually,
the elastic energy of the core becomes dominant (NA u2/NB  1). In thiscase (C3 regime), the structural parameters are obtained as c  N 2=3 g 1=3 ;
R
A

ð53Þ

1=3
L  NA1=9 NB g 1=9 b2=3 C i :ð54Þ

As in the C1 regime, the scaling expression for L̄ is similar to that of aplanar neutral polymer brush. The corona thickness decreases by theaddition of acid or salt. This dependence is stronger than that in the C1

Effects of Added Electrolytes

103

regime, because the interfacial area per corona chain does not decrease. At
1/6 21/2b  b3N ; N21/6
N3/4
C̄i, L̄ reaches the minimum size N1/2
A
B ḡ
B . Forb . b3N, neither the acid concentration nor the ionic strength affects
the structural parameters (QN regime).
The conditions for each regime are summarized in Table 2. When all the
2/3
conditions are simultaneously fulfilled for b  1, NB is restricted to N1/3
A ḡ
4/3 2/3
NB  NA for the C1 regime, and NB  NA ḡ for the C3 regime, respect1/2
 NB  NA in the C1 regime
ively. In particular, when NB satisfies N1/2
A ḡ
2/3
in the C3 regime, the transition between theand NA  NB  N4/3
A ḡ
crew-cut and the star-like micelles occurs at u  1.
STAR-LIKE MICELLES
In this section, we discuss the structural parameters of star-like micelles(u  1). In the following, four different regimes (S1, S2, S3, and QN) aredescribed.
Low Ionic Strength Limit
In the low ionic strength limit (C̄i  ac̄p), the free energy of a star-likemicelle per chain is written from Eqs. (32) and (33) as
1=3
9 1=3 2=3
NA u2
f ðuÞ ¼ NA g
1þð1 þ log f1
ci Þ
2
NB
 ðNA NB b1 C i Þ1=2 u3=2 :

ð55Þ

1/2
It takes the minimum value when C̄i . NAN22
and C̄i .
B ḡb
2 25 3 7/2
are simultaneously satisfied (see Appendix A). Notice that
NANB ḡ b
the first condition is rewritten as C̄i . ac̄p. In this case (S2 regime), thestructural parameters of a star-like micelle are given by using Eqs. (15),(35), and (A3) as c  N 2=3 g 1=3 ;
R
A

ð56Þ

L  NA1 NB3 g 1 b1 C i ;

ð57Þ

2

a  NA2 NB4 g 2 b2 C i :ð58Þ

As in the C2 regime, the scaling expression for L̄ is equal to that of theosmotic brush. When the acid concentration is increased, L̄ decreases to
its minimum size N1/2
B . In this case, the structural parameters do notchange even if the acid is further added (QN regime).
The conditions characterizing the S2 regime are summarized inTable 2. When these conditions are simultaneously satisfied, NB  NA

104

S. Mochizuki et al.

2/3
should hold. In the range of NA  NB  N4/3
, the transition between
A ḡ
the S2 and the C2 regimes occurs at u  1.

High Ionic Strength Limit
In the high ionic strength limit (C̄i  ac̄p), the micellar free energy is
given by
1=3
9 1=3 2=3
NA u 2
NB2
1þ
þ
u3 :
f ðuÞ ¼ NA g
2
NB
4NA b2 C ið59Þ

By repeating the same discussion as in the previous section, we identify
the S1, S3, and QN regimes. The structural parameters in each regime
are summarized in Table 1. The core radius increases by adding salt
or acid in the S1 regime, while it does not change in the S3 and QN
regimes. The corona thickness in the S1 and S3 regimes is given by
1/5 3/5
N3/5
R̄c with v ¼ a2/C̄i. This scaling is similar to that of aL̄  N21/5
A
B v
spherical neutral polymer brush with an effective excluded volume
parameter v. In these regimes, the corona thickness is reduced by theaddition of electrolytes, which decreases the effective excluded volume.
This dependence in the S1 regime is weaker than that in the S3 regime,
because the core radius is increased in S1, whereas it is constant in S3.
The conditions for each regime are summarized in Table 2. The S1
1/2
and NB  NA, respectively. In
and S3 regimes exist when NB  N1/2
A ḡ
1/2 1/2
particular, if NB is restricted to NA ḡ  NB  NA in S1 and NA 
2/3
in S3, the transition between crew-cut and star-like micelles
NB  N4/3
A ḡ
occurs at u  1.
REGIME DIAGRAMS
Using the results in the previous sections as summarized in Tables 1
and 2, we now construct the regime diagrams. Depending on theneutral/charged relative block length, six different types of the diagrams
are obtained (see from Fig. 2 to Fig. 7). They contain various regimes
with different scaling expressions for the structural parameters. As stated
before, we consider the case of b  1.
NB  NA1/3ḡ2/3
Because the charged B block is much shorter than the neutral A block,the free energy of the B chain is negligible compared to that of the A chain.
The core radius is determined by the competition between the elastic
energy of the core and the interfacial energy. Hence, it depends neitherEffects of Added Electrolytes

105

FIGURE 2 Regime diagram as a function of the ionic strength C̄i and the ratio of acid concentr2/3
ation to acidic dissociation constant b for NB  N1/3
. Both axes are logarithmic. The dottedA ḡ
21/3 1/3 21/2
line (Ci ¼ NA ḡ b
) divides the QN regime into two regions. The dashed line (C̄i ¼
1/2) is the boundary between the M0 (below) and M1 (above) regions.
N22
B bon the acid concentration nor on the ionic strength. In this case, only the
C2, C3, and QN regimes appear as shown in Fig. 2.
When the acid concentration is relatively low, the salt concentration
dependences of the corona thickness vary according to the ionic strength.
At very low ionic strength, L̄ is independent of the salt concentration andthe acid concentration (QN regime). At intermediate ionic strength, L̄, because the added salt promotes the dissociation of theincreases as C̄1/3
iacidic groups (C2 regime). At high ionic strength, however, L̄ decreases, because the effective excluded volume is reduced (C3 regime).
as C̄21/3
iWhen the acid concentration is very high, L̄ remains constant (QNregime). As shown by the dotted line in Fig. 2, the QN regime can be

2/3
1/2
FIGURE 3 Regime diagram for N1/3
 NB  N1/2
. Both axes are logarithmic. The meanings
A ḡ
A ḡ
of the dotted and the dashed lines are explained in Fig. 2.

106

S. Mochizuki et al.

1/2
FIGURE 4 Regime diagram for N1/2
 NB  N2/3
A ḡA ḡ. Both axes are logarithmic. On the boundary
between the C0 and S1 regimes, the structural parameters change discontinuously. The meanings ofthe dotted and the dashed lines are explained in Fig. 2.

divided into two regions in which a behaves(
2=3
NA2=9 g 2=9 b2=3 C i
ðC i  NA1=3 g 1=3 b1=2 Þ;ab1ðC i  N 1=3 g 1=3 b1=2 Þ:

ð60Þ

A

In the C2, C3, and QN regimes, the micellar free energy per chain is
f p  NA1=3 g 2=3 :ð61Þ

FIGURE 5 Regime diagram for N2/3
A ḡ  NB  NA. Both axes are logarithmic. In the M regime,
charged copolymers do not aggregate into micelles. On the boundary between the C0 and S1 regimes,
the structural parameters change discontinuously. The meanings of the dotted and the dashed lines
are explained in Fig. 2.

Effects of Added Electrolytes

107

2/3
FIGURE 6 Regime diagram for NA  NB  N4/3
. Both axes are logarithmic. The dotted and theA ḡ
25/2 22N1/2b .
dashed lines are explained in Fig. 2. The two-dot chain line represents C̄i ¼ N25/3
A
B ḡ

The free energy of an unaggregated copolymer was obtained in Eq. (31).
Using Eq. (28), the CMC is given by(ln fcmc 2

NA1=3 g 2=3  NA2=3 g þ NB5 b2 C i

2=5

NA1=3 g 2=3  NA2=3 g  NB1=5 b4=5 C i

ðC i  NB2 b1=2 Þ;ðC i  N 2 b1=2 Þ:
B

ð62Þ
1/2
The relation C̄i ¼ N22
is drawn by the dashed line in Fig. 2. The CMC
B b
in the high ionic strength limit is smaller than that in the opposite limit.
The CMC decreases with increasing the acid concentration in the low
ionic strength limit, while it increases in the opposite limit.

2/3
FIGURE 7 Regime diagram for NB  N4/3
. Both axes are logarithmic. The meanings of the
A ḡ
dashed and the two-dot chain lines are explained in Fig. 2 and Fig. 6, respectively. The dotted line
1/2
represents C̄i ¼ NAN22
.
B ḡb

108

S. Mochizuki et al.

NA1/3ḡ2/3  NB  NA1/2ḡ1/2
In addition to the C2, C3, and QN regimes, the C0 and C1 regimes also
appear as shown in Fig. 3. The QN regime is divided into two regions
characterized by the different scalings of a as shown in Eq. (60).
When the acid concentration is very low, the structural parameterschange nonmonotonically by the addition of salt. Let us consider thesequence of QN ! C2 ! C0 ! C1 ! C3. Then R̄c remains constantin the C0 regime. In thein the QN and C2 regimes and decreases as C̄22i
C1 regime, it increases as C̄2/5
i . In the C3 regime, it is constant, because
the extension of the core chain leads to an entropic loss. On the otherin the C2
hand, L̄ is constant in the QN regime. It increases as C̄1/3
iregime and as C̄i in the C0 regime. In the C1 regime, it decreases as
and has the constant value in the C3 regime.
C̄21/5i
The addition of acid causes R̄c to increase only at low acid concentration(C0 and C1 regimes). In the other regimes, added acid does not affect R̄c.
When the acid concentration is increased, L̄ decreases monotonically in the
C0, C1, C2, and C3 regimes and becomes constant in the QN regime.
The CMC in each regime is summarized in Table 3. Because the lastterm in ln fcmc is negligible in all the regimes, the lowest and the highest
values of CMC are expected in the C0 and C1 regimes, respectively.
NA1/2ḡ1/2  NB  NA2/3ḡ
As shown in Fig. 4, the regions of small b in the C0 and C1 regimes are
replaced by the S1 regime. At low acid concentration, the discontinuoustransition between the star-like (S1) micelles and the crew-cut (C0) micelles
takes place. On the transition line, the expansion ratios are given
 C0Rc NA4=11 NB8=11 g 4=11 b6=11 . 1;
S1

R

ð63Þ

C0
L
 NA2=11 NB4=11 g 2=11 b3=11 , 1:
S1
L

ð64Þ

c

2/3
1/2
TABLE 3 Critical micelle concentration for N1/3
 NB  N1/2
.
A ḡ
A ḡ

RegimeC0C1
C2, QNC3

In fcmc þ N2/3
A ḡ
24/5 22/5C̄i
2N3Bḡ22b22 C̄2i 2 N1/5
B b
2/5 24/5 22/5
24/5 22/5
N3/5
b
C̄i
2 N1/5
C̄i
B ḡ
B b
2/3
1/5 24/5 22/5
N1/3
ḡ
2
N
b
C̄
A
B
i
2/3
þ N5Bb22 C̄2i
N1/3
A ḡ
2/3
24/5 22/5
N1/3
ḡ
2 N1/5C̄i
A
B bCondition

21(C̄i  N22)
B b
21(C̄i  N22)
B b

Effects of Added Electrolytes

109

On the other hand, the transition between the S1 and C1 regimes is
continuous.
The CMC in each regime is summarized in Table 4. These different
C2
C3
QN
C1
S1
CMCs satisfy fC0
cmc , fcmc  fcmc  fcmc , fcmc , fcmc.
NA2/3ḡ  NB  NA
In Fig. 5, there exists a regime in which the micellization does not occur
25/2 1/2
1/2
ḡ b  C̄i  N22), the micellar(M regime). In the regime (N1/3
A NB
B bfree energy fp and the free energy of an isolated copolymer f1 are approximately given byf p  NA1=3 g 2=3 ;ð65Þ
2f1  NB5 b2 C i ;ð66Þ

respectively. Because fp is larger than f1, the copolymers do not form any
micelle in the M regime. Different CMCs are summarized in Table 5.
NA  NB  NA4/3ḡ2/3
As shown in Fig. 6, both the star-like and the crew-cut regimes exist in
the diagram. In the S1 regime, R̄c increases as the acid is added. Because of
the entropic penalty arising from the stretched core chains, R̄c remains
constant in the S2 and S3 regimes. On the other hand, added acid can
reduce L̄ up to the size of R̄c, and the transition between the star-likeand the crew-cut micelles occurs. As a result, the acid concentration dependence of L̄ becomes weaker as the S2 ! C2 transition takes place, while it
becomes stronger as the S3 ! C3 transition occurs. When the ionicstrength is relatively low, the micelles transform from the S1 regime to
the C2 or QN regimes without passing through the S2 regime. In thiscase, the structural parameters change discontinuously.
3 7/2
1/2
 C̄i  N22), the micellar freeIn the M regime (N2AN25
B ḡ bB benergy fp and the free energy of an unaggregated copolymer f1 are
1/2
TABLE 4 Critical micelle concentration for N1/2
 NB  N2/3
A ḡ
A ḡ.

Regime
C0
C1
C2,QN
C3
S1

ln fcmc þ N2/3
A ḡ
24/5 22/5C̄i
2N3Bḡ22b22C̄2i 2 N1/5
B b
2/5 24/5 22/5
1/5 24/5 22/5
N3/5
ḡ
b
C̄
2
N
C̄i
B
i
B b
2/3
24/5 22/5
N1/3
2 N1/5
C̄i
A ḡ
B b
2/3
N1/3
þ N5Bb22C̄2i
A ḡ
2/3
24/5 22/5
2 N1/5
C̄i
N1/3
A ḡ
B b
24/5 22/5
1/11 6/11 24/11 22/11
N4/11
N
ḡ
b
C̄i
2N1/5
C̄i
A
B
B b

Condition

21(C̄i  N22)
B b
21(C̄i  N22
b
)
B

110

S. Mochizuki et al.
TABLE 5 Critical micelle concentration for N2/3
A ḡ  NB  NA.
Regimeln fcmc þ N2/3
A ḡ

C0
C1
C2, C3
QN

24/5 22/5C̄i
2N3Bḡ22b22 C̄2i 2 N1/5
B b
2/5 24/5 22/5
1/5 24/5 22/5
N3/5
ḡ
b
C̄
2
N
C̄i
B
i
B b
2/3
24/5 22/5
N1/3
2 N1/5
C̄i
A ḡ
B b
2/3
24/5 22/5
N1/3
2 N1/5C̄i
A ḡ
B b
1/3 2/3
5 22 2NA ḡ
þNBb C̄i24/5 22/5
N4/11
N1/11
ḡ6/11b24/11 C̄22/11
2 N1/5
C̄i
A
B
i
B b

S1

Condition

1/2(C̄i  N22)
B b
22 1/2(C̄i  NB b )approximately given by2=11;
f p  NA4=11 NB1=11 g 6=11 b4=11 C i
( 2=3
NA g
ðNA2 NB5 g 3 b7=2  C i  NA1=3 NB5=2 g 1=2 bÞ;f1 2N 5 b2 CðN 1=3 N 5=2 g 1=2 b  C i  N 2 b1=2 Þ:
B

i

A

B

ð67Þ
ð68Þ

B

Because f1 is smaller than fp , the copolymers do not form any micelle inthis regime.
We summarize the different CMCs in Table 6. In the S2, S3, C2, C3,
and QN regimes, the CMC is approximately written asln fcmc  NA1=3 g 2=3  NA2=3 g ; ln fcmc :ð69Þ

In the S1 regime, the CMC is roughly8
1=5 4=5  2=5>
Ci
; ln fLcmc>< NB bln fcmc  N 4=11 N 1=11 g 6=11 b4=11 C 2=11
i
B
A
>
>
:
2=3
H
NA g ; ln fcmc

ðC i  NA5=3 NB1=2 g 5=2 b2 Þ;ðC i  NA5=3 NB1=2 g 5=2 b2 Þ:
ð70Þ

25/2 22
The condition C̄i ¼ N25/3
N1/2
bis represented by the two-dotAB ḡ
chain line in Fig. 6. These CMCs satisfy fLcmc , f*cmc , fH
cmc.

TABLE 6 Critical micelle concentration for NA  NB.
RegimeS1
S2, S3, C2, C3
QN

ln fcmc þ N2/3
A ḡ
24/5 22/5
N1/11
ḡ6/11b24/11 C̄22/11
2N1/5
C̄i
N4/11
A
B
i
B b
2/3
1/5 24/5 22/5
N1/3
ḡ
2
N
b
C̄
A
B
i
2/3
24/5 22/5
N1/3
2 N1/5
C̄i
A ḡ
B b
2/3
5 22 2
N1/3
ḡ
þN
b
C̄i
A
B

Condition

1/2(C̄i  N22
)
B b
1/2
(C̄i  N22b)
B

Effects of Added Electrolytes

111

NB  NA4/3ḡ2/3
As shown in Fig. 7, here the crew-cut micelles are not formed. In the S2and S3 regimes, L̄ decreases with increasing b. However, L̄ cannot be
1/2
smaller than N1/2
B . When L̄  NB , the ratio between L̄ and R̄c isL NA2=3 NB1=2 g 1=3  1:

Rc

ð71Þ

Hence, the copolymers do not aggregate into any crew-cut micelle. The
QN regime is divided by the dotted line in Fig. 7. The scaling expressionsof a in these two regions are


a

2

NA2 NB4 g 2 b2 C ib1

 1=2 Þ;ðCi  NA NB2 gb
2
 1=2 Þ:ðCi  NA NB gb

ð72Þ

Different CMCs are summarized in Table 6.

EFFECTS OF HYDROPHOBICITY OF CHARGED BLOCKSIn this section, we consider the situation in which the nonelectrostatic
attractions between the charged monomers are important, [i.e., t . 0 in
Eq. (21)]. One of such attractions is the hydrophobic interaction, because
the backbone of the charged block is made of hydrophobic hydrocarbons.
For the micelles composed of charged copolymers possessing a-methylgroups on the charged block, the attraction has a great effect on theirstructures (35– 37).
In Fig. 8, we plot the micellar free energy [Eq. (21)] as a function of c̄p forvarious values of the ionic strength C̄i. For C̄i ¼ 1.0  10210, the free energy
has only a single minimum that corresponds to the “collapsed” state of thecorona chain. For C̄i ¼ 1.0  1029, however, the free energy has the secondary minimum corresponding to the “stretched” state. As shown in
Fig. 9, the stretched state is determined by the balance between the stretching energy and the translational entropy of the counterions, while the collapsed state is dominated by the nonelectrostatic interaction. As the ionicstrength is further increased, the minimum at the stretched state decreases
as shown in Fig. 8(b). When the two minimum values are equal (middle
curve), a first-order transition from the collapsed state to the stretchedstate occurs. At the transition point, the corona thickness and the dissociation degree increase discontinuously, while the core radius shrinks
abruptly. In the following, we discuss how the structural parameters
depend on the electrolyte concentration by minimizing the micellar freeenergy numerically.

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S. Mochizuki et al.

FIGURE 8 Free energy of a micelle for various values of the ionic strength. The parameters are
NA ¼ 103, NB ¼ 104, ḡ ¼ 1, t ¼ 0.3, w ¼ 1, and b ¼ 40. The abscissa axis is logarithmic.

Effects of Added Salt
First, we investigate the effects of added salt on the structural parameters for fixed values of the acid concentration b. We minimize the micellar free energy numerically with respect to c̄p or a. In Fig. 10, we show thedependences of the structural parameters on the ionic strength for various
values of b. As the ionic strength is increased, the structural parameters
return to the original values after two discontinuous changes. This reentrant behavior arises due to the competition between the nonelectrostatic
attraction among the charged monomers and the osmotic pressure
induced by the mobile ions. At low ionic strength, the structural parameters do not depend on the ionic strength, because the binary attraction
overwhelms the osmotic pressure. The dissociation degree increases with theionic strength in this regime as shown in Fig. 11, and the osmotic pressure

FIGURE 9 Contributions to the micellar free energy at the first-order transition point(C̄i ¼ 1.645  1027). fs denotes the stretching energy of the copolymer, fnel the nonelectrostatic interaction between the charged monomers, fe the translational entropy of the mobile ions, and f the total
micellar free energy. The other parameters are the same as in Fig. 8. The abscissa axis is logarithmic.

Effects of Added Electrolytes

113

FIGURE 10 Reentrant changes of the structural parameters induced by added salt for three valuesof b. The parameters are NA ¼ 103, NB ¼ 104, ḡ ¼ 1, t ¼ 0.3, and w ¼ 1. Both axes are logarithmic.

becomes larger. When these two competing interactions balance eachother, the first-order transition from the collapsed micelles to the stretched
ones takes place.
In the stretched state, the corona thickness decreases with increasingC̄i. This behavior is attributed to the screening effect on the charged
corona as detailed in Appendix B. Because the repulsion between the
charged monomers decreases due to the screening effect, the core radiusincreases so that the core– corona interfacial energy per chain is
reduced. Eventually, the osmotic repulsion balances again with the nonelectrostatic attraction, and the micelles select the collapsed state.
Fig. 10 also indicates that the transition point is dependent on the valueof b. The range of C̄i in the stretched state is narrower as b increases.

FIGURE 11 The dependences of the dissociation degree on the ionic strength for two values of b.
The other parameters are the same as in Fig. 10. Both axes are logarithmic.

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S. Mochizuki et al.

Therefore, for high acid concentration, the structural parameters remainunchanged. Notice that, for b ¼ 42 in Fig. 11, the dissociation degreeincreases monotonically without any discontinuous jump.

Effects of Added Acid
We next discuss the effects of added acid on the micellar structure for
fixed values of the salt concentration defined by C̄S55C̄i 2 K̄ab. Here we
consider these effects not only under acidic condition but also underbasic condition. In Figs. 12 and 13, the structural parameters and thedissociation degree are plotted as a function of the acid concentration
for different values of the salt concentration. At very low acid concentration (achieved by the addition of base), the structural parameters arealmost constant. Here, the weakly acidic groups on the corona chainsare completely dissociated. When the dissociation degree decreases ataround b  1023 in Fig. 13, the structural parameters start to change nonmonotonically. By the addition of acid, the corona thickness increases up
to the maximum value and then shrinks as shown in Fig. 12. When theosmotic repulsion balances with the nonelectrostatic attraction, theabrupt transformation from the stretched corona to the collapsed corona
occurs. At the transition point, the higher the salt concentration is, thesmaller the changes of the structural parameters are. For high salt concentration, the structural parameters change continuously (see the case ofC̄S ¼ 1023 in Fig. 12). When b is very large, the structural parameters
are only weakly dependent on the acid concentration.

FIGURE 12 Effects of added acid on the structural parameters for three values of the salt concentration. The parameters are NA ¼ 103, NB ¼ 104, ḡ ¼ 1, t ¼ 0.3, w ¼ 1, and K̄a ¼ 1026. Both axes
are logarithmic.

Effects of Added Electrolytes

115

FIGURE 13 Effects of added acid on the dissociation degree for three values of the salt concentration.
The parameters are the same as in Fig. 12. Both axes are logarithmic.

CONCLUSIONS AND DISCUSSION
In this paper, we investigated the effects of added electrolytes on
spherical charged micelles with weakly acidic groups on the corona
chains. First, we discussed the case in which the nonelectrostatic interaction between charged monomers can be neglected compared to theion-induced repulsion (the u-solvent case). Depending on the neutral/charged relative block length, we obtained six different types of theregime diagrams. These diagrams contain various regimes characterized
by the different scaling expressions for the structural parameters. Highly
asymmetric copolymers with a very short charged block form crew-cut
micelles (see Figs. 2 and 3). The effects of added salt on the structuralparameters at low acid concentration depend on the ionic strength; thecore radius decreases with increasing the salt concentration at relatively
low ionic strength, while it increases at high ionic strength. At very
high and low ionic strength, however, the core radius is independentof the salt concentration. On the other hand, highly asymmetric copolymers with a very long charged block form star-like micelles (see Fig. 7).
In the low acid concentration regime, added salt causes the core radius
to increase and the corona thickness to decrease. These dependences
are weaker than those of the crew-cut micelles. Unlike the case of thecrew-cut micelles, there is no regime where the core radius decreases
with increasing the salt concentration. When both the acid concentrationand the ionic strength are very low, the copolymers do not aggregate. Aslong as the difference in the polymerization between the two blocks is not
too large, the charged copolymers aggregate to form star-like micelles atrelatively low acid concentration, or crew-cut micelles at high acidconcentration (see Figs. 4 to 6). In the low acid concentration regime,

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S. Mochizuki et al.

the star-like micelles change discontinuously to the crew-cut micelles by
the addition of acid.
Second, we investigated the charged micelles composed of copolymers
with a very hydrophobic charged block (the poor solvent case). In thiscase, the micellar free energy has two local minima corresponding to
the “stretched” and “collapsed” states of the corona chains. We predict a
reentrant change of the structural parameters as the salt concentration is increased at constant acid concentration. When only acid is
added, on the other hand, the nonmonotonic change of the structuralparameters is expected. The corona thickness increases up to the maximum
value and then decreases continuously, followed by a discontinuous
stretched-to-collapsed transition. These changes arise due to the competition between the hydrophobic attraction and the ion-induced repulsion.
Previously, the structure of charged micelles with weakly acidic
groups on the corona chains were investigated theoretically by Zhulina
et al. (10, 11). They discussed the formation of spherical micelles in asolvent that is poor for the neutral blocks and marginally good for the
charged blocks. For asymmetric copolymers with a very long charged
block, they found that star-like charged micelles (Sch) and star-like
quasi-neutral micelles (Sqn) are formed depending on the ionic strengthand the acid concentration. At intermediate ionic strength and low acidconcentration, the formation of the Sch micelles was predicted. Note that
the Sch micelles are equivalent to the S1 micelles in the present paper. At
high acid concentration, the copolymers aggregate into the Sqn micellesof large aggregation number. Notice that the Sqn micelles correspondto the QN micelles in our paper, but the scaling expressions for the
structural parameters are different. For example, the core radius of
the Sqn micelles depends not only on the neutral block length but also
on the charged block length. Zhulina et al. also found that the additionof salt can induce a reentrant change Sqn ! Sch ! Sqn. Unlike ourresult, the transformation Sch ! Sqn is continuous, because the nonelectrostatic interactions between the charged monomers were taken to be
repulsive in Ref. (10). Another difference is that there is no regime
where the elastic energy of the core affects the structural parameters
as predicted for the S2 and S3 micelles. This is because the nonelectrostatic repulsion prevents the corona thickness from being too small. For
asymmetric copolymers with very long neutral blocks, however, there
are regimes where the elastic energy of the core affects the structuralparameters [CC2 micelles in Ref. (10)]. This is in agreement with
our result.
The effects of added acid on the core radius of crew-cut micelles are inqualitative agreement with the experiment on the micelles composed ofhighly asymmetric poly(styrene-b-acrylic acid) (PS-PAA) copolymers witha long PS block (22, 23). As the acid concentration was increased, theEffects of Added Electrolytes

117

core size increased rapidly. In addition, the experiment found that theeffect of added acid was stronger than that of added salt. This observation
is in agreement with our analysis (see the scaling of R̄c in the C1 regime). At
high acid concentration, however, there is a disagreement. Although we
predicted the core radius to be constant at high acid concentration (C2and C3 micelles), the experiment found that the morphology of themicelles changed after the increase in the core size.
For star-like spherical micelles of poly(styrene-b-sodium acrylate) (PSPAANa) copolymers, the effects of added salt have been investigated using
the static light scattering measurement (18). They found that the aggregation number (or the core radius) increased as a function of the salt concentration, but it remained constant above ca. 0.10 M NaCl. This behavior
can be explained within our scaling analysis. Because the added salt
reduces the electrostatic repulsion among the charged chains, the coreradius becomes larger so that the core– corona interfacial energy per
chain is reduced. For high salt concentration, however, the core does
not grow due to the entropic loss of the core chains. The reentrant transition has not yet been observed in the experiment. The reason may bethat the hydrophobic attraction between PAANa monomers is muchsmaller than the electrostatic repulsion.
Recently, the aggregation behavior of poly(methyl methacrylate-bmethacrylic acid) (PMMA-b-PMAA) copolymers over the course of neutralization was investigated (35). At low neutralization degree of PMAA chains,the copolymers with a long PMAA block formed spherical micelles andlarge compound micelles (LCMs). When the neutralization degree wasincreased, LCMs was destroyed to form spherical micelles with a more
extended corona. We have reproduced this behavior qualitatively within
our numerical calculation. The corona transforms from a collapsed state
to an extended state as a result of the competition between the hydrophobic attraction and the ion-induced repulsion. The experiment also
showed that the micellar size was nearly constant at high neutralization
degree, which is in agreement with our result.
APPENDIX A: S2 REGIME
In this appendix, we derive the condition that the free energy of a starlike micelle f(u) [Eq. (55)] takes the minimum value in the low ionicstrength limit. Depending on the magnitude of C̄i, f(u) either decreases
monotonically or takes the local minimum and maximum values. The
first derivative of f(u) is(2=3 )
df ðuÞ
A
1
1 ;
ðA1Þ
 A0 u1=2
du
u5=4 þ A2 u3=4

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S. Mochizuki et al.

where3
1  1=2;
A0 ¼ ð1 þ log f1
ci ÞðNA NB b Ci Þ
2

3=2
1 þ log f1
3=4
ci
 3=4 C i ;A1 ¼
NA1=4 NB5=4 gb
2
A2 ¼

NB
:
NA

Here, we have assumed that the logarithmic term is approximately
constant.
B , 1 with B ¼ (5/8)(3/5)3/8, df(u)/du , 0 for any u. 0.
When A1 A25/8
2
25/8
B . 1 or
When A1 A2
2 7=6 5=2 4=3C i , ð5=4Þð27=5Þ1=6 ð1 þ log f1
g b;
ci Þ NA NB

ðA2Þ

f(u) takes the local minimum and maximum values. From df(u)/d u ¼0, we
haveulmin  NA5=3 NB3 g 4=3 b1 C i ;3=5

ulmax  NA1=5 NB1 g 4=5 b3=5 C i

ðA3Þ
:

ðA4Þ

Here, we have used the following condition given by calculating
d2f(u)/d u2:
ulmin , ð3A2 =5Þ1=2 , ulmax :ðA5ÞNote that u is restricted to1=3

u , u0 ¼ NA1=3 NB1=3 b1=3 C i

:

ðA6Þ

The inequality is obtained from Eq. (12). When ulmin , u0 , ulmax or 1=2
C i . NA NB2 gb

andC i . NA2 NB5 g 3 b7=2 ;ðA7Þf(u) takes the minimum value at u ¼ ulmin. Equation (A2) holds when these
conditions are simultaneously satisfied. The global minimum does
not exist when u0 . ulmax, because f(ulmin) . f(u0). To be exact, this relation
NBḡ22.3. However, this condition is not taken into
is satisfied for b  N21/3
A
account carefully, because our aim is to obtain the asymptotic expression
for the structural parameters. Thus, f(u) takes the minimum value when
the condition Eq. (A7) holds.

Effects of Added Electrolytes

119

APPENDIX B: SCREENING EFFECT
For t . 0, the addition of salt decreases the corona thickness at intermediate ionic strength. This behavior is attributed to the screening effecton the corona. In the stretched state, the micellar free energy is written asf ðcp Þ 9 4=9 10=9 2=3 2=9 1
1c p ;
 NA NBg cp þ

t
ðB1Þ
NB
2
2 2b2 C iwhere we have used c̄p  1, R̄c  L̄ and ac̄p  C̄i. Minimizing f withrespect to c̄p and using Eqs. (4) and (20), we obtain the scaling expression
3/5 1/5
N3/5
veff with veff ¼ (2b2C̄i)21 2 t. This expression indicatesL̄  N21/5
A
B R̄cthat the corona shrinks, because the added salt decreases the effective
volume veff, which is the screening effect on the charged corona.
ACKNOWLEDGMENTS
This research is partly supported by the Grant-in-Aid for Scientific
Research (C), Japan Society for the Promotion of Science under Grant
No. 15540395.
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