Theory of liquid crystalline micelles
Akihiko Matsuyama
Citation: J. Chem. Phys. 138, 034902 (2013); doi: 10.1063/1.4774386
View online: http://dx.doi.org/10.1063/1.4774386
View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v138/i3
Published by the American Institute of Physics.

Additional information on J. Chem. Phys.
Journal Homepage: http://jcp.aip.org/
Journal Information: http://jcp.aip.org/about/about_the_journal
Top downloads: http://jcp.aip.org/features/most_downloaded
Information for Authors: http://jcp.aip.org/authors

THE JOURNAL OF CHEMICAL PHYSICS 138, 034902 (2013)

Theory of liquid crystalline micelles
Akihiko Matsuyamaa)
Department of Bioscience and Bioinformatics, Faculty of Computer Science and Systems Engineering,
Kyushu Institute of Technology, Kawazu 680-4, Iizuka, Fukuoka 820-8502, Japan(Received 12 November 2012; accepted 21 December 2012; published online 16 January 2013)
A theory is introduced to describe self-assembly of liquid crystalline AB diblock copolymers, consisting of a homopolymer (A) and a side-chain liquid crystalline polymer (B). We derive the free energy of the liquid crystalline micellar solutions and examine the equilibrium solution properties: critical micelle concentration (CMC), nematic-isotropic phase transition (NIT) of the rigid side-chains
inside the micelle core, and phase separations. It is shown that there is a critical micelle size below
which the NIT becomes continuous due to a packing effect. We also find re-entrant micellizations
near the NIT temperature. The phase diagrams, including binodal, spinodal, CMC, and NIT curves
are also examined on the temperature-concentration plane. © 2013 American Institute of Physics.
[http://dx.doi.org/10.1063/1.4774386]
I. INTRODUCTION

Self-assembled polymer micelles have been studied intensively because of their potential applications in fields suchas drug delivery and biomaterial science (see reviews1, 2 ).
One of the most used methods of forming polymer micelles
is self-assembling diblock copolymers in a selective solvent for one of the two blocks. It has been reported coilcoil block copolymers,1–3 polymeric surfactants,4–6 wormlike micelles,7, 8 and rod-coil block copolymers with rigid
blocks.9–13 The stiff or rigid segments as one of the blocks
results in liquid crystalline ordering.14–18 Diblock copolymers with side-chain liquid crystalline polymers and lyotropic
chromonic liquid crystals19, 20 with a plate-like or disk-likecentral core have also attracted more attention in the recent
years.21 In these systems, it is important to understand the cooperative phenomena between self-assembly and liquid crystalline ordering.22
Block copolymers that join a random-coil polymer witha side-chain liquid crystalline block can self-assemble into avariety of morphologies: spheres, cylinders, and lamellae, andform nematic and smectic phases.23–28 The important featureis the interplay between two levels of order and self-assembly:
the liquid crystalline ordering of rigid side-chains (mesogens)inside the micelle core and of aggregates due to microphaseseparations.21 Recent experiments have showed that efficientand stable drug incorporation in polymeric micelles is feasible by the use of side-chain liquid crystalline polymers.26
Introduction of the liquid crystalline molecules to the interior of the polymeric micelle arrows for the phase transitionof the micelle inner core between the nematic phase and thesolid crystalline phase. Compare to the rigid solid state of
the core, the highly fluid character of the liquid crystalline
phase enhances incorporation of hydrophobic drugs into thecore. However, such liquid crystalline micelles, in which thea) Electronic mail: matuyama@bio.kyutech.ac.jp. URL: http://iona.bio.

kyutech.ac.jp/~aki/.
0021-9606/2013/138(3)/034902/8/$30.00

reversible self-assembled aggregates show liquid crystalline
phases, are not well-understood class of soft matter.
In this paper, we theoretically study micellar formationsof diblock copolymers, consisting of a homopolymer (A) anda side-chain liquid crystalline polymer (B) (see Fig. 1). The
spherical micelles are formed by the interaction between the
core-forming block (B) and solvent. Depending on the thermodynamic variables, nematic ordering can take place in thecore of micelles. We here focus spherical micelles and neglect
cylinders and lamellae phases. Based on the Flory-Huggins
theory for polydisperse polymer solutions,29 the Maier-Saupe
theory for nematic ordering,30 and the Tanaka theory for associating polymer solutions,31, 32 we derive the free energyof the liquid crystalline micellar solutions. From the equilibrium conditions for molecular associations, we derive thesize-distribution of the micelles of diblock copolymers andcalculate orientational order of the core of the micelles, depending on the size of the micelles. The equilibrium solution
properties, including critical micelle concentration (CMC),
nematic-isotropic phase transition (NIT), and phase separations, are discussed. We find the re-entrant micellization near
NIT temperature and various phase separations.

II. FREE ENERGY OF LIQUID CRYSTALLINEMICELLAR SOLUTIONS

We consider a binary mixture of a AB block copolymer
and a solvent molecule. Each block copolymer chain consistsof a linear homopolymer (A) and a side-chain liquid crystalline polymer (B). Let nA be the number of segments on thepolymer A and nB the number of segments on the polymer
B. The side-chain liquid crystalline polymer (B) consists ofa flexible backbone chain and rigid side-chains (mesogens).
The repeating unit on the side-chain liquid crystalline polymer is given by a backbone chain of the number nb of the segments and a mesogen with the axial ratio nm (= L/ d), where
L is the length and d is the diameter of the mesogen. The total number of segments on the side-chain liquid crystalline

138, 034902-1© 2013 American Institute of Physics

034902-2

Akihiko Matsuyama

J. Chem. Phys. 138, 034902 (2013)

and the volume fraction of the solvent molecules is
φ 0 = 1 − φ. The volume fraction of the polymer A is givenby
φA = nA∞

rNr /Nt = (nA /n)φ,

(4)

r=1

and that of the polymer B is
φB = nB∞

rNr /Nt = (nB /n)φ.

(5)

r=1

The free energy of the micellar solution can be given by32
F = Fmcl + Fmix + Fnem .

(6)

The first term Fmcl corresponds to the free energy of the micelles in the reference state where the micelles and the solvent
molecules are separately prepared and is given by
Fmcl = N0 μ◦0 +∞Nr μ◦r ,(7)

r=1

FIG. 1. Liquid crystalline micelle (r-cluster) of AB diblock copolymers,
consisted of homopolymers (A) and side-chain liquid crystalline polymers(B), where r is the number of the AB copolymer in the spherical micelle. The
core of the micelle has side-chain liquid crystalline polymers (B).

polymer is given by nB = (nm + nb )t, where t is the number of a repeating unit. Then, the total number of segmentson the AB copolymer is given by n = nA + nB . We here takethat the solvent molecule is a good solvent for the polymer A
and is a poor solvent for the polymer B. In a thermal equilibrium, the AB block copolymers aggregate into micelles,
depending on temperature and concentration. We here take
into account spherical micelles and neglect a cylindrical shapeand the other shape of the aggregates. In the following, wecall the spherical micelles as r-clusters, where r(= 1, 2, . . . ,∞) is the number of the AB copolymer in the cluster (seeFig. 1). We here call the block copolymers of r = 1 “unimers”
to avoid confusion with monomers. To derive the populationof such clusters, we consider the thermodynamics of the micellar solution.
Let a3 be the volume of an unit segment (a = d), N0 thenumber of the solvent molecules, and Nr the number of the rclusters. The total number of segments in our system is givenbyNt = N0 + n

∞


rNr ,

(1)

r=1

(2)

the volume fraction of the diblock copolymer AB is given by
φ=

∞

r=1

φr

+χA0 φA φ0 + χB0 φB φ0 + χAB φA φB ,(8)

where β = 1/kB T: T is the absolute temperature and kB is theBoltzmann constant, χ ij is the Flory-Huggins interaction parameter between the i and j components. The number of thecontact between the B segment and solvent molecule may
change upon molecular association and the term of χ B0 in
Eq. (8) may be modified. We here assume, however, thatthe same form with Flory’s model remains varied after
association.
The third term in Eq. (6) shows the nematic free energy
for aggregates and is given by
Fnem =∞


Nr Fnem,r ,

(9)

r=1

where we have taken that the number of segments of the solvent molecule is unity. The volume of the system is given by
V = a 3 Nt .
The volume fraction of the r-cluster is given by
φr = nrNr /Nt

where μ◦r is the standard chemical potential of the single isolated r-cluster and μ◦0 is that of the solvent.
The second term shows the free energy change required in the process of mixing the clusters with the solvent
molecules. According to the Flory lattice theory for polydisperse polymer solutions, the free energy of mixing is given
by29

∞

φr
ln φr
βFmix = Nt φ0 ln φ0 +
nr
r=1


(3)

where the Fnem,r is the nematic free energy for the r-cluster.
The chemical potential of the solvent molecule is given
by


∂F
βμ0 =
∂N0 Ni (i=0)1φ= βμ◦0 + ln(1 − φ) + 1 −
nr
n

nB
nA nB
A
+
χA0 +
χB0 −
φ 2 (10)
χ
AB
n
n
n2

034902-3

Akihiko Matsuyama

J. Chem. Phys. 138, 034902 (2013)

where μ◦∞ is the bulk free energy per copolymer and γ is theinterfacial tension. Then Eq. (16) is given by

and the chemical potential of the r-cluster is given by


∂F
βμr =
∂Nr Ni (i=r)
1φ= βμ◦r + ln φr + 1 − nr + nr 1 −nr+ r nA χA0 (1 − φ) + nB χB0 (1 − φ)2r = αr(r −1/3 − 1),
where

3nB
α ≡ 4π
4π

2

nA nB
+
χAB (2 − φ)φ + Fnem,r ,
n



(11)

where

(19)

1/3
(a 2 βγ )(20)

is the numerical parameter related to the interfacial tension
and depends on temperature.
B. Nematic free energy change nem,r for a r-cluster

r =

∞


rNr /

r=1

= φ/

∞


Nr

r=1

∞


(φr /r)(12)

r=1

is the average cluster size of the micelles.
To find the equilibrium distribution of clusters, we impose the multiple equilibrium conditions for r-clusters givenby32
μr = rμ1 .

(13)

Substituting Eqs. (11) into (13), we obtain the volume fractionof the r-clusters:
φr = Kr φ1r(14)

and the association constant is given by
Kr = exp[r − 1 − r − nem,r ],(15)

r ≡ β(μ◦r − rμ◦1 )

(16)

where

is the isotropic free energy change andnem,r ≡ β(Fnem,r − rFnem,1 )(17)is the nematic free energy change due to the formation of ar-cluster. In Sec. III, we derive the free energy changes for theformation of the r-cluster.

Sr =P2 (cos θ )fr (θ )d ,In this section, we derive the free energy changes r andnem,r for the formation of the r-cluster.
A. Isotropic free energy change r for a r-cluster

We first consider the isotropic free energy change r .
A spherical aggregate with r copolymers has a radius
R = (3rv/4π )1/3 , where v is the volume of a side-chain liquid crystalline polymer (B) in the core of the r-cluster and isgiven by v = a 3 nB . The chemical potential of the r-cluster isgiven by33
(18)

(21)

where d = 2π sin θ dθ , P2 (cos θ ) ≡ (3/2)(cos 2 θ − 1/3), andfr (θ ) is the orientational distribution function of mesogens inside the core of the r-cluster.
The nematic free energy of the r-cluster is given by
 3
xm
R
fr (θ ) ln 4πfr (θ )dβFnem,r =a3
nm1− νxm2 Sr2 + βFanc ,2(22)where ν(≡ U0 /kB T) is the orientational-dependent anisotropic(Maier-Saupe) interaction parameter between the mesogens30
and Fanc is the anchoring free energy of the mesogens inside
the core. The value (R/a)3 = rnB , corresponds to the total
number of polymer segments inside the core. The free energy change Fanc corresponds to the anchoring free energy ofmesogens on the surface of a micelle core and is scaled as theanchoring free energy density (fa ) times a surface area ( dS (R/a)2 ):35, 36
βFanc = xm βfa

III. FREE ENERGY OF LIQUID
CRYSTALLINE MICELLESμ◦r = rμ◦∞ + 4π R 2 γ ,We here consider the nematic free energy change nem,rdue to the formation of the r-cluster. Inside the core of the micelle, the mesogens can be oriented depending on the temperature and the concentration. Let xm = nm /(nm + nb ) be the volume fraction of mesogens inside the core. The orientational
order parameter of mesogens inside the core of the r-cluster
is given by34(n · c)2 dS

1= xm ω (2Sr + 1)(R/a)2 ,3(23)

where n is a local director of mesogen, c is an easy director,
and fa is the anchoring free energy of a mesogen on the coresurface. We here assume that the free energy change fa due tomicellizations mainly comes from the entropic loss (sa < 0)to pack a mesogen inside the core: fa = −Tsa . We here callit “packing effects.” We then have the numerical parameterω(≡ −sa /kB > 0) related to the packing entropy of mesogens inside the core.
The orientational distribution function fr (θ ) of the rcluster is determined by minimizing the free energy (22) withrespect to this function: (δFnem,r /δfr (θ ))fi (θ) = 0. We then

034902-4

Akihiko Matsuyama

J. Chem. Phys. 138, 034902 (2013)

obtainIV. PHASE SEPARATIONS

fr (θ ) =1
exp[Ar P2 (cos θ )],
Zr

(24)

where we define2
Ar ≡ nm νxm Sr + ω(rnB )−1/3 .
3(25)

The value of Ar means the strength of a nematic ordering andthe larger values of ω, xm , and ν promote nematic ordering.
The first term shows the attractive interaction between mesogens and the second term corresponds to the packing effectsof mesogens. When r → ∞, the second term becomes zeroand the value of Ar results in the mesogens in the bulk. The
constant Zr is determined by the normalization condition forthe distribution function
1

= 1,

(26)

Zr = 4π I0 [Ar ],(27)

fr (θ )d0

aswhere the function I0 [Ar ] is defined as
1

Iq [Ar ] ≡[P2 (cos θ )]q exp[Ar P2 (cos θ )]d(cos θ )

(28)

0

q = 0, 1, 2, . . . . Substituting Eq. (24) into (21), we obtain the self-consistency equation for the orientational orderparameter Sr :Sr = I1 [Ar ]/I0 [Ar ].

(29)

Substituting Eq. (24) into (22), we obtain the nematic freeenergy of the r-cluster
βFnem,r = rnB

1 2 2 xmln(I0 [Ar ])νx S −
2 m r
nm

1− ωxm (rnB )−1/3
3

where we defineBr ≡ nB(34)

where F is the free energy of the micellar solutions (Eq. (6))
andF ◦ = Nt (μ◦0 φ0 + μ◦1 φ).

(35)

is the reference free energy before the mixing. According to
the Gibbs-Duhem equation, the free energy of our system isgiven by


∞

μr
φr .
F = Nt μ0 φ0 +
(36)
nr
r=1
Using the multiple equilibrium conditions (Eq. (13)), the freeenergy F is given by

μ1 
φ .
(37)
F = Nt μ0 φ0 +nSubstituting Eqs. (35) and (37) into (34), we obtainμ1F = Nt μ0 φ0 +
φ ,
n

(38)

where μ0 ≡ μ0 − μ◦0 and μ1 ≡ μ1 − μ◦1 . Then the freeenergy for the mixing is given by substituting Eqs. (10) and(11) into (38):


1
φ
1
1−
φ
βF /Nt = φ0 ln φ0 + ln φ1 +
n
n
r

+ φB
(30)

(31)



 xm
1 2 2
I0 [Ar ]2lnνx S − S1 −2 m r
nm
I0 [A1 ]1− ωxm (nB )−1/3 (r −1/3 − 1) ,3

F = F − F ◦+ χA0 φA φ0 + χB0 φB φ0 + χAB φA φB

and the nematic free energy change nem,r (Eq. (17)) is givenbynem,r = rBr ,In this section we derive the free energy change dueto the mixing of solvent molecules and AB copolymers(monomers). The free energy for the mixing is given by32

(32)

as s function of the aggregation number r.
Substituting Eqs. (19) and (31) into (15), we obtain thevolume fraction of the r-cluster:1(33)
φr = [φ1 exp(α + 1 − αr −1/3 − Br )]r ,e
as a function of the volume fraction of an unimer φ 1 andthe aggregation number r. The total volume fraction of thecopolymer in the system is given by Eq. (3).

1 2 2 xm
νx S −ln(I0 [A1 ])2 m 1 nm1
− ωxm (nB )−1/3 ,3(39)

and we here define the free energy per a segment:f ≡ F/Nt .
The spinodal line which separates stable and unstable regions is obtained by (∂ 2 f/∂φ 2 )T = 0, or equivalently from(∂μ0 /∂φ)T = 0.38, 39 This leads to


1
nA
nB
nA nB
χA0 +
χB0 −
2
χ
AB −
2
n
n
n
1−φ


φ ∂r
1
2−
= 0.
(40)
−
nrφ
r2 ∂φThe condition for the phase separations in a thermal equilibrium is given by μ0 (φ ) = μ0 (φ ) and μr (φ ) = μr (φ ) for r= 1, 2, . . . , where φ and φ are the copolymer concentrations
in lower and higher concentration phase, respectively. Using
the multiple equilibria conditions (Eq. (13)), the coexistence
curves (binodals) of the phase separations are derived by the034902-5

Akihiko Matsuyama

J. Chem. Phys. 138, 034902 (2013)

μ0 (φ ) = μ0 (φ ),(41)

μ1 (φ ) = μ1 (φ ).

(42)

The binodal lines can also be derived by a double tangent
method where the equilibrium concentrations fall on the same
tangent line to the free energy curve (Eq. (39)).
The osmotic pressure  is related to the solvent’s chemical potential by a3 β = −βμ0 . In the dilute concentrations,
we can expand μ0 in power series of the volume fraction φand we obtain


1
1
3
+
− χ (χij ) φ + · · · ,(43)a β/φ =nr
2where χ (χ ij ) is the coefficient of φ 2 in Eq. (10). The aggregation number r can be measured from the intercept, plottedagainst φ. In Sec. V, we show some numerical results.
V. NUMERICAL RESULTS

In our numerical calculations, we take nm = nb = 3,
t = 30, and ω = 0.3 for a typical example: nA = 100, nB= 180. Using a temperature parameter τ , we can write ν = 1/τ
and α = α 0 /τ , where we put α 0 = 4. On increasing α, or decreasing temperature, the association constant increases andwe have larger aggregates.
A. Orientational order parameters of r-clusters

By numerically solving Eq. (29), we can obtain the orientation order parameter Sr of the inner core of the r-cluster as afunction of temperature. Figure 2 shows the orientational order parameter Sr for the r-clusters plotted against the temperature, where TNI shows the NIT temperature of the mesogens
in the bulk state. We find the critical micelle size Mc , or a critical aggregation number of copolymers, for the NIT curves.
For the large micelles with r > Mc , we have the first-order
NIT and for small micelles with r < Mc we have the continuous NIT, due to the packing effect of mesogens. These results
are consistent with the recent experimental results.40 The behavior is the same as NIT in surface-aligned nematic films41
and the transition becomes second order at a critical thickness.

Order parameter Sr

0.8
0.6
0.4

103

0.2

106

r=1
10
102

108

0.0
0.8

0.9

1.0

1.1

1.2

1.3

T / TNI
FIG. 2. Orientation order parameter Sr of the inner core of the r-cluster as afunction of temperature. The temperature TNI is the NIT temperature of themesogens in the bulk state (r = ∞).

Order parameter Srcoupled equations
0.6
0.4
0.1
0.08
0.06

0.05

0.2

0.04
0.02
b=0

0.0
0.20

0.21

0.22

0.23
1/u

0.24

0.25

FIG. 3. Orientational order parameter Sr as a function of u−1 for various
values of b. A critical point (closed circle) appears for bc = 0.05, where thesecond-order NIT takes place.

The packing effect of the mesogens inside the small micelles
changes from the first-order NIT to the continuous one. On
decreasing the size of the micelles, the NIT temperature increases and the change of the order parameter becomes continuous. In the small micelles, the mesogens inside the core
have a paranematic (or weak nematic) state with Sr = 0, evenat high temperatures.
We here put Ar = uSr + b in Eq. (25), in which we define
the constant u(≡ nm xm ν) for the anisotropic interaction term
between mesogens and b(≡ (2/3)nm ω(rnB )−1/3 ) for the packing effects. Figure 3 shows the orientational order parameterSr as a function of u−1 for various values of b. When b = 0
the first-order NIT takes place at 1/u = 0.223.34, 37 We find a
critical point (closed circle) for bc = 0.05, where the secondorder NIT takes place. Then the critical micelle size for the
NIT is given by
Mc =

1
nB



2nm ω
0.15

3
,

(44)

and increases with (nm ω)3 . In Fig. 2, we have Mc = 9.6. Notethat the volume per a core with the critical micelle size isgiven by a3 (Mc nB ). When b < bc , or r > Mc , we have thefirst-order NIT, but for b > bc , or r < Mc , the order parameter
continuously changes.

B. CMC and NIT curves

In the following, we consider only unimers (r = 1) andmicelles (r = M) with aggregation number M: φ = φ 1 + φ M ,because the summation of the infinite series (3) cannot numerically converge. We here take the aggregation number M= 100, in which all micelles are taken to have the same aggregation number M. As shown in Fig. 2, the NIT temperature is
T/TNI  1.06 for M = 100. Figure 4 shows the volume fractions of unimers and micelles with M = 100, as a functionof the volume fraction φ of the copolymer for T/TNI = 0.84.
Below the CMC, almost the copolymer is present as unimers.
Above the CMC, as extra copolymer is added to the solution
almost goes into micelles. As shown in Fig. 2, at this temperature, the mesogens inside the core of micelles and unimers
are oriented and show a nematic phase with SM  S1  0.7.
The value of the order parameter does not depend on theAkihiko Matsuyama

J. Chem. Phys. 138, 034902 (2013)

Volume fraction of micelles

0.008
T/TNI=0.84

0.004
φ1

0.002

CMC

0

0

φΜ
0.005
φ

0.01

FIG. 4. Volume fractions of unimers (φ 1 ) and micelles with M = 100 (φ M )as a function of the volume fraction φ of the copolymer for T/TNI = 0.84.

concentration φ, because the mesogens inside the core of micelles do not interact with the mesogens on the other micelles.
Figure 5 shows the CMC curve (dashed-curve) and the
NIT line (horizontal dotted-line) on the (φ, T/TNI ) plane. We
here define the NIT temperature, where the order parameter
SM of the micelle jumps. We find four different regions. The
region (1) shows that the isotropic (or paranematic) unimers
exist. In the region (2), we have the nematic unimers, in which
the core B of the unimers is in a nematic state. From the regions (1) to (2), the order parameter S1 continuously increases
with decreasing temperature. In the region (3), the isotropic
micelles exist and the nematic micelles appear in the region(4), where the core of the micelles is in a nematic state. From
the regions (3) to (4), the order parameter SM discontinuously increases at the NIT temperature as decreasing temperature. We also find the re-entrant micellization for 0.006 < φ< 0.008, where the system changes form (1) → (3) → (2)→ (4) with decreasing temperature. The order parameter S1of the unimer sharply increases around T/TNI  1.15, where
the isotropic micelles (3) break to nematic unimers (2) andthe CMC curve shifts to higher concentrations. The nematicordering of unimers leads to the retardation of micellization.
Figure 6 shows the volume fractions of unimers (solidline) and micelles (dotted-line) with M = 100 as a function oftemperature at φ = 0.004 (a), φ = 0.007 (b), and φ = 0.012(c) in Fig. 5. In the dilute concentrations (a), the nematic micelles grow up at a critical micelle temperature and the volume fraction of the nematic unimers decreases. In Fig. 6(b),

T/TNI

1.4

(1)

1.2

(3)

1.0
(2)

0.8

0

(4)

0.005

0.010
φ

0.015

0.020

FIG. 5. CMC curve (dashed-curve) and NIT line (horizontal dotted-line) onthe (φ, T/TNI ) plane. We have four different regions: isotropic unimers (1),nematic unimers (2), isotropic micelles (3), and nematic micelles (4) regions.
The re-entrant micellization takes place at 0.006 < φ < 0.008 with decreasing
temperature.

Volume fraction of micelles

0.006

Volume fraction of micelles

Volume fraction of micelles

034902-6

0.004(a) φ =0.004

φ1

0.003
0.002
0.001

φ 100
0
0.7

0.8

0.9

1.0

1.1

1.2

1.3
(b)

0.006

φ =0.007
0.004
0.002
0
0.8
0.012

0.9

1.0

1.1

1.2

1.3

φ =0.012
(c)

0.008

0.004

0
0.8

0.9

1.0

1.1 1.2
T/TNI

1.3

1.4

FIG. 6. Volume fractions of unimers (φ 1 : solid line) and micelles (φ M :dotted-line) with M = 100, as a function of temperature at φ = 0.004 (a),φ = 0.007 (b), and φ = 0.012 (c) in Fig. 5.

we have the re-entrant micellization at two critical micelletemperatures. The volume fraction of the isotropic micelles
first increases at T/TNI = 1.23. With decreasing temperature,
the isotropic micelles break up to nematic unimers and thevolume fraction φ M goes to zero. Further decreasing temperature, we have the second micellization at T/TNI = 0.99, where
the nematic unimers aggregate and form nematic micelles. In
Fig. 6(c), the micellization takes place at T/TNI = 1.38 andthe volume fraction φ M increases. With decreasing temperature, the isotropic unimers change to nematic unimers near
T/TNI = 1.15, where the volume fraction φ M of the isotropic
micelles decreases. Further decreasing temperature, the volume fractions φ 1 and φ M jump at the NIT temperature: T/TNI= 1.05, where the isotropic micelles change to the nematic
micelles.
These CMC and NIT curves may be hidden inside coexisting curves, depending on the strength of the interactions
between solvents and copolymers. In Sec. V C, we calculate
phase separations including the CMC and NIT curves on thetemperature-concentration plane.

C. Phase separations

In this subsection, we calculate binodal and spinodal
curves on the temperature-concentration plane. We here

034902-7

Akihiko Matsuyama

J. Chem. Phys. 138, 034902 (2013)

1.3

(3)

(1)

T/TNI

1.2
1.1
1.0
(2)

0.9
(4)

0.8

0

0.02

φ

0.04

0.06

FIG. 7. Phase diagram on the (φ, T/TNI ) plane for c1 = 0.5. The solid curveshows the binodal (coexistence) and the dotted-curve is the spinodal curve.
The closed circle corresponds to the critical point. The phase diagram has the
CMC curve (dashed-curve) and NIT line (horizontal dotted-line), appearedin Fig. 5.

assume that the solvent molecule is a good solvent for thehomopolymer (A): χ A0 = 0, and put χ ≡ χ B0 = χ AB . Theinteraction parameter χ is proportional to the inverse temperature and then the Flory-Huggins interaction parameter isgiven by χ = c1 ν = c1 /τ , where c1 is the numerical parameter.
Figure 7 shows the phase diagram with an upper critical solution temperature (UCST) for c1 = 0.5 with nA = 100. The
solid curve shows the binodal (coexistence) and the dottedcurve is the spinodal curve. The closed circle corresponds
to the critical point. Between the binodal and spinodal lines,
we have a metastable region and below the spinodal line, wehave an unstable region. Above the NIT temperature, twophase coexistence between two isotropic micelle phases appears in the region (3). Below the NIT temperature, we have
two-phase coexistence between two nematic micelle phasesin the region (4). Note that the binodal and spinodal lines athigher concentrations are not depicted. For example, at T/TNI= 0.9, the spinodal concentration exists at φ = 0.12.
Figure 8 shows the spinodal curves for various values of
c1 . On increasing the value of c1 , the interaction between the
copolymer B and the solvent molecule becomes poorer. We
find that the critical point shifts to higher temperatures andthe CMC line is hidden inside binodal and spinodal lines, inwhich we have two phase coexistence between an isotropic
unimer phase in the region (1) and an isotropic micelle phasein the region (3) above the NIT temperature. Below the NIT,

1.3

T/TNI

1.0

1.1

0.45

1.0

0.432

0.8

In this paper we present a mean field theory to describe
self-assembly of liquid crystalline AB diblock copolymers,
consisting of a homopolymer (A) and a side-chain liquid crystalline polymer (B). We derive the free energy of the micellar solutions and calculate the orientational order parameterof mesogens inside the core of micelles, depending on thesize of micelles. It is shown that there is a critical size Mcof the micelle: large spherical micelles with r > Mc have afirst-order NIT, but small micelles with r < Mc have a continuous NIT due to the packing effects of mesogens inside thecore. The value of Mc is proportional to (nm ω)3 . We also findthe re-entrant micellization near the NIT temperature and the
UCST type phase diagrams which have four different regions:isotropic unimer, isotropic micelle, nematic unimer, and nematic micelle regions. The cooperative phenomena between
self-assembly and nematic ordering of the micelle can be controlled by the temperature, concentration, and the mesogensof side-chain liquid crystalline polymers.
In this paper we focus on dilute concentrations and consider spherical micelles. At higher concentrations, cylindricalor lamellar aggregates may appear. The theory presented here
can be the basis for the further studies.

0.5

1.2

0.9

VI. SUMMARYACKNOWLEDGMENTS

(3)

(1)

two phase coexistence between a nematic unimer phase (2)and a nematic micelle phase (4). On the other hand, as decreasing c1 , the critical point shifts to lower temperatures below the NIT temperature. When c1  0.432, the critical point
meets the NIT line and the intersection becomes a tricriticalpoint.
In this paper, we have assumed only two-distributionof M-clusters and unimers. Recent experiments have shownthat the PEG-b-P(AzoPyl) micellar solution indicates that two
different-size of micelles are distributed in the solution.26 In
this case we can consider three-distribution model, including
unimers, M1 -clusters, and M2 -clusters. However the results
are qualitatively similar with Fig. 5. The volume fractions of
micelles increase at CMC and the amount of larger micelles
increases with increasing the polymer concentration.
In our model, if we take nA = 0, nb = 0, and t = 0,
the theory can describe the spherical aggregates of a lowmolecular-weight liquid crystal (nB = nm ) dispersed in poor
solvent molecules.

c1=0.43

(2)

0.02

φ

1 G. Riess, Prog. Polym. Sci. 28, 1107 (2003).
2 A. Blanazs, S. P. Armes, and A. J. Ryan, Macromol. Rapid Commun. 30,(4)0This work was supported by Grant-in Aid for Scientific
Research (C) (Grant No. 23540477) from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

267 (2009).

0.04

0.06

3 S. Jain and F. S. Bates, Science 300, 460 (2003).
4 A. L. R. Bug, M. E. Cates, S. A. Safran, and T. A. Witten, J. Chem. Phys.

87, 1824 (1987).
FIG. 8. Spinodal curves (dotted-curves) on the (φ, T/TNI ) plane for various
values of c1 . The closed circle shows the critical point. The phase diagram
has the CMC curve (dashed-curve) and NIT line (horizontal dotted-line), appeared in Fig. 5.

5 R. Nagarajan, J. Chem. Phys. 90, 1980 (1989).
6 Z. G. Wang and S. A. Safran, J. Chem. Phys. 89, 5323 (1988).
7 Soft and Fragile Matter, edited by M. E. Cates and M. R. Evans (Instituteof Physics, Bristol, 2000).

034902-8

Akihiko Matsuyama

8 M. E. Cates, J. Phys. (France) 49, 1593 (1988).
9 N. R. Scruggs, R. Verduzco, D. Uhrig, W. Khan, S.-Y. Park, J. Lal, and J.

A. Kornfield, Macromolecules 42, 299 (2009).
10 S.-Y. Park, T. Kavitha, T. Kamal, W. Khan, T. Shin, and B. Seong, Macromolecules 45, 6168 (2012).
11 L. Jia, P-A. Albouy, A. Di Cicco, A. Cao, and M.-H. Li, Polymer 52, 2565
(2011).
12 S. V. Ahir, A. R. Tajbakhsh, and E. M. Terentjev, Adv. Funct. Mater. 16,556 (2006).
13 P. van der Schoot and M. E. Cates, Europhys. Lett. 25, 515 (1994).
14 M. Muller and M. Schick, Macromolecules 29, 8900 (1996).
15 R. Holyst and M. Schick, J. Chem. Phys. 96, 721 (1992).
16 M. Matsen, J. Chem. Phys. 104, 7758 (1996).
17 R. R. Netz and M. Schick, Phys. Rev. Lett. 77, 302 (1996).
18 D. R. M. Williams and G. H. Fredrickson, Macromolecues 25, 3561 (1992).
19 D. J. Edwards, J. W. Jones, O. Lozman, A. P. Ormerod, M. Sintyureva, and
G. J. T. Tiddy, J. Phys. Chem. B 112, 14628 (2008).
20 J. Lyden, Liq. Cryst. 38, 1663 (2011) and see references cited therein.
21 Q. Li, Liquid Crystals Beyond Displays (Wiley, New Jersey, 2012).
22 M. P. Taylor and J. Herzfeld, J. Phys.: Condens. Matter 5, 2651 (1993).
23 M. Anthamatten, J. S. Wu, and P. T. Hammond, Macromolecules 34, 8574
(2001).
24 W. Y. Zheng and P. T. Hammond, Macromolecules 31, 711 (1998).
25 I. W. Hamley, V. Castelletto, Z. B. Lu, C. T. Imrie, T. Itoh, and M. AliHussein, Macromolecules 37, 4798 (2004).

J. Chem. Phys. 138, 034902 (2013)
26 M. Nishihara, Y. Murakami, T. Shinoda, J. Yamamoto, and M. Yokoyama,

Chem. Lett. 37, 1214 (2008).
27 K. Okano, Y. Mikami, and T. Yamashita, Adv. Funct. Mater. 19, 3804

(2009).
28 H. Yu, T. Kobayashi, and H. Yang, Adv. Mater. 23, 3337 (2011).
29 P. J. Flory, Principles of Polymer Chemistry (Cornell University, Ithaca,1953).
30 W. Maier and A. Saupe, Z. Naturforsch. 14a, 882 (1959).
31 A. Matsuyama and F. Tanaka, Phys. Rev. Lett. 65, 341 (1990).
32 F. Tanaka, Polymer Physics (Cambridge University Press, Cambridge,2011).
33 J. N. Israelachvili, Intermolecular and Surface Forces (Academic, Boston,1992).
34 P. G. de Gennes and J. Prost, The Physics of Liquid Crystals (Oxford Uni-

versity Press, New York, 1993).
35 G. Barbero and G. Durand, in Liquid Crystals in Complex Geometries,

edited by G. P. Crawford and S. Zumer (Taylor & Francis, London, 1996),
Chap. 2.
36 R. W. Ruhwandl and E. M. Terentjev, Phys. Rev. E 56, 5561 (1997).
37 A. Matsuyama and T. Kato, J. Chem. Phys. 105, 1654 (1996).
38 C. Shen and T. Kyu, J. Chem. Phys. 102, 556 (1995).
39 A. Matsuyama, in Encyclopedia of Polymer Blends, Volume 1: Fundamentals, edited by A. I. Isayev (Wiley-VCH, Weinheim, 2010), Chap. 2.
40 S. Bono and J. Yamamoto, private communication (September 2012).
41 P. Sheng, Phys. Rev. Lett. 37, 1059 (1976).