Phase diagrams of binary mixtures of liquid crystals and rodlike polymers in
the presence of an external fieldAkihiko Matsuyama and Tomomi Ueda
Citation: J. Chem. Phys. 136, 224904 (2012); doi: 10.1063/1.4728337
View online: http://dx.doi.org/10.1063/1.4728337
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THE JOURNAL OF CHEMICAL PHYSICS 136, 224904 (2012)

Phase diagrams of binary mixtures of liquid crystals
and rodlike polymers in the presence of an external fieldAkihiko Matsuyamaa) and Tomomi Ueda
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 26 April 2012; accepted 24 May 2012; published online 12 June 2012)
We theoretically study phase separations in mixtures of a low molecular-weight-liquid crystalline
molecule (LC) and a rigid-rodlike polymer (rod) under an external field, such as magnetic or electricfields. By taking into account two orientational order parameters of the rod and the LC, we define
four nematic phases (N0 , N1 , N2 , N3 ) on the temperature-concentration plane. Depending on the signof the dielectric anisotropy  i of the rod (i = 1) and LC(i = 2), we examine the phase behavior of
rod/LC mixtures in the case of  1 > 0,  2 > 0 (a),  1 < 0,  2 > 0 (b),  1 > 0,  2 < 0 (c),and  1 < 0,  2 < 0 (d). We predict a variety of phase separations induced by an external field.
© 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4728337]
I. INTRODUCTION

Alignment of rigid-rodlike polymers in liquid crystalline solvents has been the current topic in physics
and material science.1–14 Recently, we have presented amean field theory to describe phase separations in binary
mixtures of a low-molecular weight liquid crystal (LC)and a long rigid-rodlike polymer (rod),15 including carbon
nanotube,16–18 Kevler: a rigid-rodlike polymer of poly(pphenylene terephthalamide),19 tobacco mosaic virus,20 andliquid crystalline polymers. By taking into account two orientational order parameters of the rod and the LC, we havepredicted three uniaxial nematic phases (N0 , N1 , N2 ) on thetemperature-concentration plane, depending on the attractive or repulsive interactions between the rod and the LC.
Biaxial nematic phases in rod/LC mixtures have also been
predicted.21 It is well known that such rods and LCs can bereoriented in the presence of an external field such as electricor magnetic fields.22–27
In this paper, we consider the phase behavior of rod/LC
mixtures in the presence of the external fields. When the external magnetic or electric field applies to the system (parallel to z axis), we can expect four types of uniaxial nematic
phases. Figure 1 schematically shows the four uniaxial nematic phases, defined by using the orientational order parameter (S1 ) of the rod and that (S2 ) of the LC. In this paper,
we consider uniaxial nematic phases. The nematic N0 phaseshows the rods and the LCs are parallel to the field: S1 > 0
and S2 > 0. The nematic N1 phase is defined as that the rods
are perpendicular to the external field and the LCs are parallelto the field: S1 < 0 and S2 > 0. In this phase, the rods are randomly distributed within the perpendicular plane to the external field (note that there is no positional order for rods alongz axis). The nematic N2 phase is defined as that the rods areparallel to the external field and the LCs are perpendicular
to the field: S1 > 0 and S2 < 0. The LCs are randomly distributed within the perpendicular plane to the external field.

The nematic N3 phase is defined as that the rods and LCs are
randomly distributed within the perpendicular plane to the external field: S1 < 0 and S2 < 0. These nematic phases can beformed by depending on the positive or negative dielectric (ordiamagnetic) anisotropy  i of the rod (i = 1) and LC(i = 2).
For example, when the dielectric anisotropy  1 of the rods is
positive and the  2 of LCs is negative, we can expect the N2
phase in the presence of a strong electric field. Moreover, ina weak external field, the interaction between the rod and LC
can prevail the system. We then expect that the phase behavior of the mixtures can be changed by the coupling between
the dielectric anisotropy and the strength of the external field.
In order to predict theoretically the phase diagrams of therod/LC mixtures in the presence of an external field, we extend the previous model15 to this system by taking into account the interaction between an external field and each component. Depending on the interaction between the rod and LC
and the strength of the external field, we find a variety of phaseseparations.

II. THEORY

We here briefly introduce the free energy of mixtures ofa rod and a LC.15 We consider a binary mixture of N1 rigidrodlike polymers (rods) of length L1 and diameter D1 and N2
low-molecular weight LC molecules of length L2 and diameter D2 : L1  L2 . The volume of the rod and that of the
LC are given by v1 = (π/4)D12 L1 and v2 = (π/4)D22 L2 , respectively, where V = v1 N1 + v2 N2 . Let φ1 = v1 N1 /V and
φ2 = v2 N2 /V be the volume fraction of the rod and LC, respectively. Using the axial ratio n1 ≡ L1 /D1 of the rod andn2 ≡ L2 /D2 of the LC, the volume per particle is given by
v1 = a 3 n1 and v2 = a 3 n2 , where we assume D ≡ (D1 = D2 )and define a3 ≡ (π /4)D3 .
The free energy F of the rod/LC mixture is given by15
F = Fmix + Fnem + Fext .

a) URL: http://iona.bio.kyutech.ac.jp/~aki/.

0021-9606/2012/136(22)/224904/8/$30.00

136, 224904-1(1)© 2012 American Institute of Physics

224904-2

A. Matsuyama and T. Ueda
Z

J. Chem. Phys. 136, 224904 (2012)

interaction between LCs can be negligible because the lengthof LC is short.
The orientational order parameter Si (i = 1, 2) of a nematic phase is given by22Si = P2 (cos θ )fi (θ )d ,
(4)

Z

N0

N1

Z

Z

N2

N3

E

FIG. 1. Schematically illustrated four possible uniaxial nematic phases in
the presence of the external field, parallel to z axis. The nematic phases aredefined by using the orientational order parameter S1 of the rod and that S2of the LC: nematic N0 phase with S1 > 0 and S2 > 0, nematic N1 phase with
S1 < 0 and S2 > 0, nematic N2 phase with S1 > 0 and S2 < 0, and N3 phasewith S1 < 0 and S2 < 0.

where fi (θ ) is the distribution function of the particle i(= 1, 2)
and P2 (cos θ ) ≡ 3(cos 2 θ − 1/3)/2.
When ν 12 > 0 in Eq. (3), LCs prefer along align parallel to rods: S1 > 0 and S2 > 0, to be lower than the nematic
free energy. This corresponds to the N0 and N3 phases. On
the other hand, when ν 12 < 0, LCs prefer along align perpendicular to rods to be lower than the nematic free energy. This
corresponds to either N1 or N2 phase.
The last term in Eq. (1) shows the free energy change
due to an external field such as electric or magnetic fields. Let 1 ≡  1,  −  1, ⊥ be a dielectric (or diamagnetic) anisotropyof the rod and  2 ≡  2,  −  2, ⊥ be a dielectric anisotropyof the LC. When the external electric field E is applied to thesystem, the free energy is given by22, 33, 34a 3 βFext /Nt = −φ1 β1 (n · E)2 f1 (θ )d− φ2 β2(l · E)2 f2 (θ )d= − φ1 β1 E

The first term shows the free energy for mixing of rods
and LCs in the isotropic phase
φ1
φ2
a βFmix /V =ln φ1 +
ln φ2 + χ φ2 φ1 ,
n1
n2
3

1
1
− (ν1 + 5/4)φ12 S12 − ν2 φ22 S22
2
2− ν12 φ1 φ2 S1 S2 ,where d= 2π sin θ dθ , ν 1 (≡ U1 /kB T > 0) is theanisotropic attractive (Maier-Saupe) interaction parameter between rods,30, 31 and ν 2 (≡ U2 /kB T > 0) is the anisotropic
attractive interaction between LCs. The anisotropic interaction ν 12 (≡ U12 /kB T) between a LC and a rod can be positive or negative value. The term ν 1 and 5/4 correspond to
the attractive and excluded volume interactions between rods,
respectively.15, 32 We here assume that the excluded volume

cos2 θf2 (θ )d1= − [φ1 h1 (1 + 2S1 ) + φ2 h2 (1 + 2S2 )],3(5)
where n and l are the unit orientation vector of the rod andthat of the LC, respectively. We here define the dimensionlessexternal field parameters: h1 ≡  1 βE2 and h2 ≡  2 βE2 .
Depending on the strength of the external field and the positive or negative dielectric anisotropy of the two components,
we have four uniaxial nematic phases.
In a thermal equilibrium state, the distribution function
f1 (θ ) and f2 (θ ) are determined by minimizing the free energy(1) with respect to these functions: (δF /δf1 (θ )){f2 (θ)} = 0 and(δF /δf2 (θ )){f1 (θ)} = 0. We then obtain1
exp[ 1 P2 (cos θ )],4π I0 [ 1 ]

(6)



2
(ν1 + 5/4)φ1 S1 + ν12 φ2 S2 + h1 ,3(7)

f1 (θ ) =(3)

cos2 θf1 (θ )d− φ2 β2 E 2(2)where β ≡ 1/kB T; T is the absolute temperature, kB is theBoltzmann constant. The first and the second terms correspond to the entropy of isotropic mixing for rods and LCs,
respectively, and the last term shows the Flory-Huggins interaction parameter χ (≡ U0 /kB T) between a rod and a LC, where
U0 is the interaction energy between the rod and the LC in an
isotropic state.28 A positive χ denotes that the rod-LC contacts are less favored compared with the rod-rod and LC-LC
contacts.
The second term in Eq. (1) shows the free energy for nematic ordering29
 φi 
fi (θ ) ln 4πfi (θ )da 3 βFnem /V =ni=1,2 i2

where
1 ≡ n1and the function I0 [ 1 ] is defined by 1[P2 (cos θ )]m exp[xP2 (cos θ )]d(cos θ ),Im [x] ≡(8)0m = 0,1,2. Substituting Eq. (6) into Eq. (4), we obtainthe self-consistency equation for the orientational order224904-3

A. Matsuyama and T. Ueda

J. Chem. Phys. 136, 224904 (2012)

parameter S1

1.10

where we define

1
exp[ 2 P2 (cos θ )],4π I0 [ 2 ]


2
ν
h
≡
n
φ
S
+
ν
φ
S
+
2
2
2 2 2
12 1 1
2 .
3

(10)

−
0.90 N
0.0
1.10
I
1.05

(11)

T / TNI

Substituting Eq. (10) into Eq. (4), we obtain the selfconsistency equation for the orientational order parameter S1
(12)

N0
1.00

0.2

0.4

0.6

0.8

1.0
(c)

1.00
N2

TCP

(13)

N1
0.80
0.0

0.2

0.4

φ1

0.6

0.8

1.0

FIG. 2. Typical phase diagrams for c12 = 0 (a), c12 = 0.5 (b), and c12= −0.3 (c) without the external field: h1 = h2 = 0.

(14)

(15)

Using the reduced temperature τ , the interaction parameteris given by ν 2 = 4.55/(n2 τ ). The value of n2 is on the order of one and then the parameter ν 2 remains finite values.
We here define the anisotropic attractive interaction parameter between LCs
α = ν2 /χ ,

1.0

0.8

0.90

◦
where TNIshows the NIT temperature of a pure liquid crystal.
We here define the reduced temperature◦τ ≡ T /TNI
= 4.55/(n2 ν2 ).

0.6

IThen the total free energy (1) is given by the sum of Eqs. (2),(5), and (13).
When φ 1 = 0, or pure LCs, the nematic-isotropic phasetransitions (NIT) take place at22, 32◦
n2 ν2 = n2 U2 /(kB TNI) = 4.55,

0.4

(b)

0.90
0.0
1.10

T / TNI1(ν1 + 5/4)φ12 S122
1+ ν2 φ22 S22 + ν12 φ1 φ1 S1 S2
2
φ1
φ2− ln I0 [ 1 ] −ln I0 [ 2 ].
n1
n2

0.2

0.95

By numerically solving the coupled equations (9) and (12), we
can obtain the values of the nematic order parameters as afunction of temperature and concentration.
Substituting Eqs. (6) and (10) into Eq. (3), the free energy
Fnem of the nematic ordering is given bya 3 βFnem /V =

I

1.00
0.95



S2 = I1 [ 2 ]/I0 [ 2 ].

+
N

1.05

Similarly, we obtainf2 (θ ) =(a)

(9)
T / TNIS1 = I1 [ 1 ]/I0 [ 1 ].

(16)

where we take that LCs are good solvents for rods: χ  1.
When χ = 0 in a theta solvent, we can remove the parameter α from numerical calculations. The nematic parameter αmeans that the ratio between the nematic interaction ν 2 between LCs and the unfavorable interaction χ between LC androd in an isotropic phase. On decreasing the value of α, the unfavorable interaction becomes dominant in the free energy.35
The anisotropic attractive interaction parameter between rods
is defined as
c1 = ν1 /ν2 ,(17)

c12 = ν12 /ν2 ,(18)

and we also define

for the anisotropic interaction parameter between a rod anda LC. The parameters c1 and c12 are on the order of one fordispersion forces. The value of c12 becomes important in thephase behavior. The parallel or perpendicular alignments in
rod/LC mixtures can be modified by wrapping polymers or
surfactants on rod’s surface.36, 37 Hence, it can control the balance between the attractive van der Waals interaction and excluded volume between a LC and a rod. These modifyings cancontrol the interaction parameter c12 in our model.
III. RESULTS

In this section, we show phase diagrams calculated fromthe free energy. The coexistence (binodal) curve can be obtained by solving the chemical potential balances between
two separated phases and can also be derived by a double tangent method where the equilibrium volume fractions fall on
the same tangent line to the free energy curve.15 In order tocalculate phase diagrams, we set n1 = 10, n2 = 1, α = 10,
and c1 = 0 in the following.
A. Phase diagrams in the absence of an external field

Before we calculate the phase diagrams under the external fields, such as magnetic or electric fields, it is important to224904-4

A. Matsuyama and T. Ueda

J. Chem. Phys. 136, 224904 (2012)order parameters

1.0

1.10

S1
T / TNI

S2

0.2

0.4

0.6

0.2

0.4

0.6

0.8

1.05

0.5
S2
0.0

1.00

N0

0.95

(b)
0.2

0.4

0.6

0.90
0.0
1.10

0.8

0.2

0.4

0.6

0.8

1.0
(c)

1.05

S1
T / TNIorder parameters

1.0

0.5
S2

0.0

0.2

0.4

(c)

φ1

0.6

N1

N2

1.00
0.95

S2
S1

–0.5

1.0
(b)

S1
T / TNIorder parameters

1.00

0.90
0.0
1.10

0.8

1.0

–0.5

+
N0

0.95

(a)
–0.5

(a)

1.05

0.5

0.0

−
N0

0.90
0.0

0.2

0.4

0.8

FIG. 3. Order parameters for c12 = 0 (a), c12 = 0.5 (b), and c12 = −0.3(c) at τ = 1.05 without the external field: h1 = h2 = 0. The two verticalbroken lines denote the isotropic (I) and nematic binodals in Fig. 2.

show the typical phase diagrams in the absence of an external
field. We here show three typical phase diagrams. Refer to theprevious paper15 for details.
Figure 2 shows the phase diagrams for c12 = 0 (a), c12= 0.5 (b), and c12 = −0.3 (c) in the absence of the external field: h1 = h2 = 0. The solid curve shows the binodal.
The dotted line shows the first-order NIT and the dashed line
shows the second-order NIT. Figure 3 shows the order parameters for c12 = 0 (a), c12 = 0.5 (b), and c12 = −0.3 (c) at
τ = 1.05 in Fig. 2. As shown in Fig. 3(a), in the N+ phase,the rods are oriented and the LCs are randomly distributedwith S2 = 0. At the N0 phase [Fig. 3(b)], both the rods and
LCs are in a nematic phase. In Fig. 3(c), the isotropic phase iscontinuously changed to the N1 phase at φ 1  0.31. Further
increasing concentration, we have the first-order N1 -N2 phasetransition at φ 1  0.42.
B. Phase diagrams in the presence of an external field

In this subsection, we show some numerical results of thephase diagrams under an external field. Figures 2(a)–2(c) are
drastically changed by the external field. Depending on the

φ1

0.6

0.8

1.0

FIG. 4. Phase diagrams for c12 = 0 (a), c12 = 0.5 (b), and c12 = −0.3(c) with the weak external field: h1 = h2 = 0.1.

sign of the dielectric anisotropy  i (i = 1, 2), we have thefollowing four cases as discussed in Fig. 1.
1. When  1 > 0 and  2 > 0

Figure 4 shows the phase diagrams under the weak external field: h1 = h2 = 0.1. When  1 > 0 and  2 > 0, therods and LCs favor to align parallel to the direction of the external field and we have S1 > 0 and S2 > 0: N0 phase, undera strong external field. When c12 = 0 (Fig. 4(a)), the isotropic
phase appeared in Fig. 2(a) disappears and we have two phase+coexistence (N−0 +N0 ) between two paranematic N0 phaseswith different concentrations. When c12 = 0.5 with the attractive interaction between a rod and a LC (Fig. 4(b)), thephase separation I+N0 disappears and we have the stableparanematic N0 phase. For c12 = −0.3 (Fig. 4(c)), we havethe paranematic N1 phase at low concentrations, where mostLCs align parallel to the direction of the external field androds are perpendicular to the LCs due to the repulsive interaction c12 (<0). On increasing the concentration of rods,
the excluded volume between rods prevails the system andthe paranematic N2 phase appears. The isotropic phase appeared in Fig. 2(c) disappears due to the external field and224904-5

A. Matsuyama and T. Ueda

J. Chem. Phys. 136, 224904 (2012)

1.10

1.10
(a)

1.00

1.05
T / TNIT / TNI1.05− +N0 + N0

0.95
0.90
0.0
1.10

0.2

0.4

0.6

0.8

T / TNIT / TNI

0.4

0.6

0.8

1.0

1.05
− +
N0 + N0

N0

1.00
0.95

0.2

0.4

0.6

0.8

0.90
0.0
1.10

1.0

(c)

1.05
− +
N0 + N 0

1.00
0.95

0.2

0.4

φ1

0.6

0.8

0.2

0.4

0.6

0.8

1.0
(c)

1.05
T / TNIT / TNI

0.2

(b)

0.95

0.90
0.0

N0
1.00

0.90
0.0
1.10

1.0

1.05

0.90
0.0
1.10

(a)
N1

0.95

(b)

1.00

−

N1

N2

1.00

N2

0.95

1.0

0.90
0.0

0.2

0.4

φ1

0.6

0.8

1.0

FIG. 5. Phase diagrams for c12 = 0 (a), c12 = 0.5 (b), and c12 = −0.3(c) with the strong external field: h1 = h2 = 1.

FIG. 6. Phase diagrams for c12 = 0 (a), c12 = 0.5 (b), and c12 = −0.3(c) with the weak external field: h1 = −0.1 and h2 = 0.1.

the N1 and N2 phases become stable under the weak external field, where the intermolecular interaction c12 prevails thesystem.
On increasing the strength of the external field hi , the N1+
and N2 phases disappear and we have the N−
0 +N0 phase separation under the strong external field h1 = 1.0 (see Fig. 5).
The stable N0 phase under a weak external field in Fig. 4(b)is separated into two N0 phases (Fig. 5(b)). We find that thestrong external field induces the phase separations between a−
rod-rich N+
0 phase and a rod-poor N0 phase.

On increasing the concentration of rods, the excluded volumeinteractions between rods prevail the mixtures and we have+the N0 phase. We also have the triple point (N−1 +N1 +N0 ) at
τ  0.97. Above the triple point, we have the phase separa+tions: N1 +N0 and N−
1 +N1 . When c12 = 0.5 (Fig. 6(b)), theisotropic phase appeared in Fig. 2(b) shifts to higher temperatures due to the external field and we have the stale paranematic N0 phase. When c12 = −0.3 (Fig. 6(c)), the I-N1 phasetransition appeared in Fig. 2(c) disappears and we have theN1 -N2 phase transition. On increasing the concentration ofrods, the excluded volume interactions between rods prevail
the system and the rods align parallel to the direction of theexternal field and the LCs align perpendicular to the rods because of the repulsive interaction c12 < 0. We then have theparanematic N2 phase at high concentrations. We also have+the triple point (N−1 +N1 +N2 ) at τ  0.96. On increasing
the strength of the external field hi , the N0 and N2 phases in+Fig. 6 change to the N1 phase and we have the N−
1 +N1 phaseseparation, as shown in Fig. 5.

2. When  1 < 0 and  2 > 0

Figure 6 shows the phase diagrams under the weak external field: h1 = −0.1 and h2 = 0.1. When  1 < 0 and 2 > 0, the rods (LCs) favor to align perpendicular (parallel) to the direction of the external field and we have S1 < 0and S2 > 0: N1 phase, under a strong external field. Whenc12 = 0 (Fig. 6(a)), the isotropic phase disappears and wehave the first-order N1 -N0 phase transition. In the paranematic
N1 phase at low concentrations, the rods with the negative
anisotropy ( 1 < 0) are randomly distributed on the plane
perpendicular to the external field and the LCs with the positive anisotropy ( 2 > 0) align parallel to the external field.

3. When  1 > 0 and  2 < 0

Figure 7 shows the phase diagrams under the weak external field: h1 = 0.1 and h2 = −0.1, for  1 > 0 and  2 < 0. In

224904-6

A. Matsuyama and T. Ueda

J. Chem. Phys. 136, 224904 (2012)

1.1
N2

(a)

1.0

0.9

0.9

N0
0.8
0.0
1.10

N1
0.8
0.0
1.1

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0
(b)

(b)

N3
1.05

N3
T / TNIT / TNIN2

N3

T / TNIT / TNI

−

1.0

1.1

(a)
+
N2

N0

1.00

1.0

N0

0.95
0.90
0.0
1.10

0.2

0.4

0.6

0.8

0.9
0.0
1.1

1.0

0.2

0.4

0.6

0.8
(c)

(c)
−

+

N2

T / TNIT / TNI

1.05 N2
1.00

N3

1.0

N2

N2
0.9

0.95
N1
0.90
0.0

1.0

0.2

0.4

φ1

0.6

0.8

1.0

FIG. 7. Phase diagrams for c12 = 0 (a), c12 = 0.5 (b), and c12 = −0.3(c) with the external field: h1 = 0.1 and h2 = −0.1.

this case, the rods (LCs) favor to align parallel (perpendicular)
to the direction of the external field and we have S1 > 0 and
S2 < 0: N2 phase, under a strong external field. When c12 = 0(Fig. 7(a)), the isotropic phase disappears and we have thefirst-order N0 -N2 phase transition. In the paranematic N2
phase, the rods which have the positive dielectric anisotropy( 1 > 0) align parallel to the external field and the LCs
with the negative anisotropy ( 2 < 0) are randomly distributed on the plane perpendicular to the external field. At
low temperatures and low concentrations, however, the attractive interaction between LCs prevails the system and then theparanematic N0 phase appears. We also have the triple point+(N0 +N−
2 +N2 ) at τ  0.9. Above the triple point, we have−+the phase separations: N0 +N−
2 and N2 +N2 . When c12 = 0.5(Fig. 7(b)), the isotropic phase appeared in Fig. 2(b) shifts tohigher temperatures due to the external field and the paranematic N3 phase appears. At low concentrations, the LCs with
the negative dielectric anisotropy align perpendicular to theexternal field and the rods align parallel to the LCs due tothe attractive interaction (c12 > 0) between the rod and LC
and then we have the N3 phase. The external field is not sostrong to form the N2 phase. On increasing the concentrationof rods, the excluded volume interaction between rods prevails the system and the stable paranematic N0 phase appears.

N1
0.8
0.0
0.2

0.4

φ1

0.6

0.8

1.0

FIG. 8. Phase diagrams for c12 = 0 (a), c12 = 0.5 (b), and c12 = −0.3(c) with the external field: h1 = h2 = −0.1.

When c12 = −0.3 (Fig. 7(c)), the I-N1 phase transition appeared in Fig. 2(c) disappears and we have the N1 -N2 phase+transition. We also have the triple point (N−1 +N1 +N2 ) at
τ  0.91. On increasing the strength of the external field hi ,the N0 , N1 , and N2 phases in Fig. 7 change to the N1 phase+and we have the N−
1 +N1 phase separation under the strong
external field.
4. When  1 < 0 and  2 < 0

Figure 8 shows the phase diagrams under the weak external field: h1 = h2 = −0.1. When  1 < 0 and  2
< 0, the rods and LCs favor to align perpendicular to the direction of the external field and we have S1 < 0 and S2 < 0:N3 phase, under a strong external field. When c12 = 0(Fig. 8(a)), the isotropic phase disappears and we have thefirst-order N3 -N2 , N1 -N3 phase transitions. In the paranematic
N3 phase, the rods and LCs with the negative anisotropy ( i< 0) are randomly distributed on the plane perpendicular to
the external field. At lower temperatures, however, the attractive interaction between LCs prevails and the LCs alignparallel to the external field and the paranematic N1 phase appears. On increasing the concentration of rods, the excluded
volume between rods prevails the system and the rods align224904-7

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J. Chem. Phys. 136, 224904 (2012)

FIG. 9. Possible nematic phases depending on the interaction parameter c12 and the external field |h| for  1 > 0,  2 > 0 (a),  1 < 0,  2 > 0 (b),  1
> 0,  2 < 0 (c), and  1 < 0,  2 < 0 (d). We here take |h| ≡ |h1 | = |h2 |. Symbols show the possible nematic phases on the (|h|, c12 ) plane. For example,
the symbol • shows that the N1 or N2 phase appears on the temperature-concentration plane. Figure 4(c) corresponds to the point (|h| = 0.1, c12 = −0.3) inFig. 9(a).

parallel to the external field. We then have the paranematic
N2 phase. We also have the triple point (N1 +N3 +N2 ) at τ
 0.86. Above the triple point, we have the phase separations: N1 +N3 and N3 +N2 . When c12 = 0.5 (Fig. 8(b)), theisotropic phase appeared in Fig. 2(b) shifts to higher temperatures due to the external field and the paranematic N3
phase appears. On increasing the concentration of rods, theexcluded volume interaction between rods prevails the system and the paranematic N0 phase appears. When c12 = −0.3
(Fig. 8(c)), the I-N1 phase transition appeared in Fig. 2(c) disappears and we have the first-order N1 -N2 , N3 -N2 , and thesecond-order N2 -N3 phase transitions. We also have the triple
point (N1 +N3 +N2 ) at τ  0.85. On increasing the strengthof the external field hi , the N0 , N1 , and N2 phases in Fig. 8+changes to the N1 phase and we have the N−
3 +N3 phase separation in the presence of the strong external field.
C. Possible nematic phases

As shown in Subsections III A and III B, the interaction
parameter c12 and the strength of the external field are important to understand the phase behavior. Figure 9 shows thepossible nematic phases on (|h|, c12 ) plane for  1 > 0,  2
> 0 (a),  1 < 0,  2 > 0 (b),  1 > 0,  2 < 0 (c), and  1
< 0,  2 < 0 (d). We here take |h| ≡ |h1 | = |h2 |. Symbols
on Fig. 9 show the possible nematic phases on the (|h|, c12 )plane. For example, the symbol • at (|h| = 0.1, c12 = −0.3) inFig. 9(a) means that the N1 and N2 phases, as shown in
Fig. 4(c), appear on the temperature-concentration plane. The
symbol ◦ at (0.1, 0.5) in Fig. 9(b) means that the N0 phaseappears as shown in Fig. 6(b). Figures 4(a)–4(c) correspondto the point (0.1, 0.0), (0.1, 0.5), and (0.1, −0.3) on Fig. 9(a),respectively. Figures 5(a)–5(c) correspond to the point (1.0,0.0), (1.0, 0.5), and (1.0, −0.3) on Fig. 9(a), respectively.
Figures 6(a)–6(c) correspond to the point (0.1, 0.0), (0.1, 0.5),
and (0.1, −0.3) on Fig. 9(b), respectively. Figures 7(a)–7(c)correspond to the point (0.1, 0.0), (0.1, 0.5), and (0.1, −0.3)
on Fig. 9(c), respectively. Figures 8(a)–8(c) correspond to thepoint (0.1, 0.0), (0.1, 0.5), and (0.1, −0.3) on Fig. 9(d), respectively. When  2 > 0 (Figs. 9(a) and 9(b)), the N0 or N1
phase appears for c12 > 0 in the weak external field. On increasing the external field, the field prevails the system andwe have the N0 phase for  1 > 0,  2 > 0 (a) and the N1
phase for  1 < 0,  2 > 0 (b). When  2 < 0 (Figs. 9(c)and 9(d)), the N0 , N2 , or N3 phase appears for c12 > 0 in the
weak external field. On increasing the external field, the field
prevails the system and we have the N2 phase for  1 > 0,
 2 < 0 (c) and the N3 phase for  1 < 0,  2 < 0 (d). We
find various nematic phases depending on the strength of theexternal field and the interaction c12 .
IV. SUMMARY

We have theoretically studied the phase diagrams of
rod/LC mixtures under an external field. By taking into account two orientational order parameters of the rod and LC,
we predict four possible nematic phases (N0 , N1 , N2 , N3 ) on
the temperature-concentration plane. Depending on the signof the dielectric (or diamagnetic) anisotropy  i of the rod(i = 1) and LC(i = 2), we examine the phase behavior for
rod/LC mixtures. For example, when the dielectric or diamagnetic anisotropy of rods is positive ( 1 > 0) and thatof LCs is negative ( 2 < 0), the phase behavior is summarized in Fig. 9(c) on the (|h|, c12 ) plane. In the weak external

224904-8

A. Matsuyama and T. Ueda

field, the interaction between molecules prevails the system
and we have the variety phase behavior including N0 , N1 , N2 ,and N3 , depending on the sign of the anisotropy  i . In thestrong external field, the effect of the external field enhances+
the phase separations and we have the broad N−
0 +N0 phase−+separations for  1 > 0,  2 > 0, N1 +N1 for  1 < 0,  2+−+
> 0, N−
2 +N2 for  1 > 0,  2 < 0, and N3 +N3 for  1
< 0,  2 < 0. We hope our results encourage further experimental and theoretical studies of phase separations of dispersions of rods (nanotubes) in nematic LCs. These results canbe useful to obtain the desired director orientation during thephase separation process and new materials with orientational
order in rod/LC mixtures .
ACKNOWLEDGMENTS

This 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.
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