Theory of polymer-dispersed cholesteric liquid crystals
Akihiko Matsuyama
Citation: J. Chem. Phys. 139, 174906 (2013); doi: 10.1063/1.4828940
View online: http://dx.doi.org/10.1063/1.4828940
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THE JOURNAL OF CHEMICAL PHYSICS 139, 174906 (2013)

Theory of polymer-dispersed cholesteric liquid crystals
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 5 August 2013; accepted 22 October 2013; published online 7 November 2013)
A mean field theory is presented to describe cholesteric phases in mixtures of a polymer and acholesteric liquid crystal. Taking into account an anisotropic coupling between a polymer and a liquid
crystal, we examine the helical pitch, twist elastic constant, and phase separations. Analytical expressions of the helical pitch of a cholesteric phase and the twist elastic constant are derived as a functionof the orientational order parameters of a polymer and a liquid crystal and two intermolecular interaction parameters. We also find isotropic-cholesteric, cholesteric-cholesteric phase separations, andpolymer-induced cholesteric phase on the temperature-concentration plane. We demonstrate that an
anisotropic coupling between a polymer and a liquid crystal can stabilize a cholesteric phase in the
mixtures. Our theory can also apply to mixtures of a nematic liquid crystal and a chiral dopant. We
discuss the helical twisting power, which depends on temperature, concentration, and orientational
order parameters. It is shown that our theory can qualitatively explain experimental observations.
© 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4828940]
I. INTRODUCTION

In a common nematic phase, the nematic director n ishomogeneous through a system, although, a cholesteric liquid crystal phase is a chiral nematic: the director n is uniformly twisted along a perpendicular axis called helical axis.
The distance along the helical axis for the director to twist 2π
is called pitch p = 2π /q, which depends on temperature andconcentration in a system. Since p is comparable to an optical wavelength, the periodicity results in Bragg scattering oflight beams. This mesoscopic helical structure becomes thebasis of optoelectronic applications such as a twisted nematic
liquid crystal display.1, 2
Cholesteric liquid crystals consist of either chiralmolecules which are optically active molecules or nematic
molecules with chiral dopants.3–5 At small concentration ofa chiral dopant, the pitch is inversely proportional to theconcentration of the chiral dopant.3 Such cholesteric phasecan be stabilized or modified by polymer networks.1, 6 It
is known as the polymer-stabilized cholesteric liquid crystals for electro-optical displays.7–10 The pitch and the physical properties of cholesterics can be controlled by mixing
cholesteric liquid crystals and polymer chains, or photo reactive molecules.2, 11–13
It has been shown both experimentally14–17 andtheoretically18–25 that in mixtures of a polymer and a liquid
crystal (so-called polymer dispersed liquid crystals: PDLCs),
phase separations between an isotropic and a nematic ora smectic phase appears below the nematic-isotropic phasetransition (NIT) temperature of the pure liquid crystal.2 For
strong anisotropic (attractive) interactions between polymers
and liquid crystals, it has been recognized that a smectic phasecan be induced in a nematic liquid crystal mixture.26–28 Sucha) Electronic

mail: matuyama@bio.kyutech.ac.jp. URL: http://iona.bio.kyutech.ac.jp/~aki/.

0021-9606/2013/139(17)/174906/10/$30.00

induced nematic phases have also been predicted in polymerliquid crystal mixtures.29 However, theoretical attempts forcholesteric phases in the mixtures are not numerous.30–33
To describe cholesteric phases, the chiral dispersion
intermolecular interactions have been treated by many
authors.30–48 The configuration of the constituent ith molecule
is characterized by its position vector ri and its orientation
vector i . It has been shown that the intermolecular chiralinteraction is given by the term (1 × 2 · r̂12 )P1 (1 · 2 )in the lowest order, where r̂12 = (r1 − r2 )/|r1 − r2 |. The chiral interaction must be odd in 1 · 2 , while the nematic interaction be even. The appearance of this term distinguishes
cholesterics from nematics and determines the cholestericpitch in an equilibrium state. Lin-Liu et al. have taken into
account the chiral potential in a series of the Legendre polynomials, truncated by P3 (1 · 2 ).40 They found that the pitch
depends on temperature through a ratio of orientational orderparameters in the form [P4 (x)/P2 (x)]2 . We here treat the chiralfree energy in the lowest order.
The aim of this paper is to develop a mean field theory
to describe cholesteric phases in mixtures of a liquid crystal and a polymer chain. Developing the molecular theory of
the cholesteric phase presented by Lin-Liu et al.,40–42 we derive the free energy of mixtures of a chiral liquid crystal anda polymer chain. Our main original contribution is to taking
into account the chiral coupling between a polymer and a liquid crystal in the lowest order. As the results, we obtain analytical expressions of the pitch of the cholesteric phase andthe twist elastic constant as a function of orientational orderparameters of a polymer and a liquid crystal. We demonstratethat the anisotropic coupling between a polymer and a liquid
crystal can stabilize the cholesteric phase in the mixtures. We
also find phase separations between isotropic-cholesteric andcholesteric-cholesteric phases. Our theory can also apply to
mixtures of a non-chiral nematic liquid crystal and a chiralmolecule. It is well known that a small amount of a chiral139, 174906-1© 2013 AIP Publishing LLC

174906-2

Akihiko Matsuyama

J. Chem. Phys. 139, 174906 (2013)

dopant in a non-chiral nematic liquid crystal phase induces acholesteric phase with a helical pitch, which is inversely proportional to the concentration of the chiral dopant molecules.3
We find that the chiral coupling between a liquid crystal and achiral dopant creates a helical twisting power, which is given
as a function of the concentration of chiral dopants and orientational order parameters.
In Sec. II, we extend a mean field theory to describe binary mixtures of a cholesteric liquid crystal and a polymer
chain. In Sec. III, we show some numerical results of pitch,
twist elastic constant, and phase diagrams for the mixtures.
In Sec. IV, pitch and helical twisting power in mixtures ofa non-chiral nematic liquid crystal and a chiral molecule are
discussed.
II. FREE ENERGY OF MIXTURES OF A POLYMER AND
A CHOLESTERIC LIQUID CRYSTAL
A. Chiral free energy

Consider a binary mixture of a liquid crystal moleculeand a polymer chain for which a nematic ordering is forbidden in the constituent pure polymer chains. To describe
cholesteric phases of the mixtures, we develop a mean fieldmodel proposed by Lin-Liu et al.40–42 We here take into account an anisotropic coupling between polymers and liquid
crystals.
Let NP is the number of a polymer chain with nP segments and NL be the number of a low-molecular weight liquid
crystal molecule of length L and diameter D. The volume of aliquid crystal and that of a polymer chain are given by
vL = (π/4)D 2 L and vP = a 3 nP , respectively. Let
φL = vL NL /V and φP = vP NP /V be the volume fraction of the liquid crystal and the polymer, respectively, where
V is the volume of the system: φ L + φ P = 1. Using the axial
ratio nL = L/D of the liquid crystal, the volume per liquid
crystal molecule is given by vL = a 3 nL , where we define
a3 = (π /4)D3 .
The free energy consists of the following two terms:
F = Fmix + Fani .

(1)

The first term in Eq. (1) is the free energy of mixing of apolymer and a liquid crystal molecule and is given by FloryHuggins theory for polymer solutions:49
φL
φP
a 3 βFmix /V =ln φL +
ln φP + χ φL φP ,
(2)
nL
nP
where χ is the Flory-Huggins interaction parameter between
a liquid crystal and a polymer in an isotropic phase and β≡ 1/kB T; T is the absolute temperature, kB is the Boltzmann
constant.
The second term in Eq. (1) shows the free energy
for cholesteric phases. The configuration of a constituent
molecule is characterized by its position vector r and its orientation unit vector , defined by a polar angle θ and an azimuthal angle ϕ, or solid angle d( = sin θ dθ dϕ), in a fixed
coordinate frame. Let fi (n(r) · ) be the orientational distribution function of a constituent molecule i(= L, P), wheren(r) is the local director. It should be noted that the distribution function depends only on the relative angle between

the local director n(r) and the molecular orientation vector .
The anisotropic part of the free energy in the second virial
approximation can be given by
ρi fi (n(r) · ) ln 4πfi (n(r) · )drd
βFani /V =
i=L,P

+


1 
ρi ρj fi (r1 , 1 )fj (r2 , 2 )2 i,j =L,P

×Uij (r1 , 1 ; r2 , 2 )dR,

(3)

where dR ≡ dr1 dr2 d1 2 and ρi = Ni /V is the number
density. The first term in Eq. (3) shows the entropy changes
due to orientational ordering and Uij is the orientationdependent intermolecular potential between two particles iand j (i, j = L, P). We here take UPP = 0 because we consider
non-nematic polymer chains in the constituent pure polymers.
The lowest-order contributions to the interaction potential forthe cholesteric phase are given by in a series of the Legendre
polynomials:40
Uij (r1 , 1 ; r2 , 2 ) = Uij,1 (r12 )(1 × 2 · r̂12 )P1 (1 · 2 )+Uij,2 (r12 )P2 (1 · 2 ),(4)

where r12 = r1 − r2 and we have truncated by P2 (x). Lin-Liu
et al. have taken into account the intermolecular potential ina series of the Legendre polynomials by P4 (1 · 2 ),40 however there are some unknown numerical parameters related
to the interaction. Equation (4) is useful to reduce the number of the unknown numerical parameters. The potential Uij, 1shows the chiral interaction between two particles i and j. The
term 1 × 2 · r̂12 represents scalars coupling between orientational and spatial variables. The potential Uij, 2 shows theintermolecular potential that accounts for the formation of anematic phase, which has been used in Maier-Saupe50 andOnsager models.51 Following the symmetry consideration forcholesteric phases, we require that the first term in Eq. (4) is
odd in 1 · 2 and the second term be even.
Substituting Eq. (4) into (3), the anisotropic free energy
can be expressed as
ρi fi (n(r) · ) ln 4πfi (n(r) · )drd
βFani /V =i=L,P+β(F1 + F2 )/V ,(5)

where we have the chiral component of the free energy1 
βF1 /V =ρi ρj fi (n(r1 ) · 1 )fj (n(r2 ) · 2 )2 i,j =L,P
× βUij,1 (r12 )(1 × 2 · r̂12 )P1 (1 · 2 )dR, (6)and the usual nematic free energy1 
βF2 /V =ρi ρj fi (n(r1 ) · 1 )fj (n(r2 ) · 2 )2 i,j =L,P
× βUij,2 (r12 )P2 (1 · 2 )dR.

(7)

The degree of local orientational order for a givenfi (n(r) · ) can be specified by the second rank order parameter tensor Q(i)αβ (r) of the molecule i and in general is a function

174906-3

Akihiko Matsuyama

of position:

J. Chem. Phys. 139, 174906 (2013)





3
1
dfi (n(r) · )α β − δαβ
2
2


3
1
≡ α β − δαβ ,22where we define×(1 × 2 · r̂12 )P1 (1 · 2 )d1 d2

and

4
(i)
(j )
αβγ (rα /r)Qμβ (r1 )Qμγ (r2 ),
9

(10)


fi (n(r1 ) · 1 )fj (n(r2 ) · 2 )P2 (1 · 2 )d1 d2
=

2 (i)
(j )
Qαβ (r1 )Qαβ (r2 ),3(11)

respectively, where αβγ is Levi-Civita antisymmetric tensor
of the third rank and r ≡ |r12 |.
(j )We here assume that the Qαβ (r2 ) does not change appreciably over the range of the potential and can be expanded in
the Taylor series about r12 :Qαβ (r2 ) = Qαβ (r1 ) + rκ∂Qαβ (r1 )∂rκ∂ 2 Qαβ (r1 )1+ rκ rλ.
2∂rκ ∂rλβUij,2 (r12 )dr12 ,(15)βUij,2 (r12 )z2 dr12 ,

(16)

z2
dr12 ,
r

(17)


(8)

where α is the Cartesian component of the molecularorientation vector  = (sin θ cos ϕ, sin θ sin ϕ, cos θ ) at a
position r.
Using Eq. (8), the averages over 1 and 2 in Eqs. (6)and (7) yield(9)fi (n(r1 ) · 1 )fj (n(r2 ) · 2 )

=


Aij =

Q(i)
αβ (r) ≡(12)

Substituting Eq. (12) into Eqs. (10) and (11), the free energy (Eq. (6)) is given by

1 2
4
(L)
(L)
βF1 /V = ρL CLL
αβγ Qμβ (r1 )∂α Qμγ (r1 )dr1
2
9

4
(L)
(P )
+ρL ρP CLP
αβγ Qμβ (r1 )∂α Qμγ (r1 )dr1 ,9(13)
and the free energy (Eq. (7)) is given by

1 2
2 (L)
Q (r1 )Q(L)βF2 /V = ρL ALL
αβ (r1 )dr1
2
3 αβ

2 (L)
)
Q (r1 )Q(P+ρL ρP ALP
αβ (r1 )dr1
3 αβ

1
1 (L)
Q (r1 )∂μ ∂μ Q(L)+ ρL2 BLLαβ (r1 )dr1
2
3 αβ

1 (L)
)
Q (r1 )∂μ ∂μ Q(P+ρL ρP BLP
αβ (r1 )dr1 ,
3 αβ
(14)

Bij =

Cij =

βUij,1 (r12 )and we have assumed that the distance between the centersof two particles is small z along the z axis. This expression is
now in the form of the integral of a potential energy. We here
employ a simple square well interaction potential with a shortrange d0 , which is the order of the particle size.40 With thisapproximation we can take
Aij = −vij νij ,(18)

1
Bij = − vij νij d02 ,3(19)

Cij = −vij cij d0 ,(20)

where vij is the average volume between the particles i and j
in random orientations: vLL = (π/4)L2 D and vLP = (π/4)
LD 2 nP . The interaction parameter ν LL (≡−ULL, 2 /kB T > 0) corresponds to the orientational-dependent(Maier-Saupe) interaction parameter between liquid crystal
molecules for a nematic phase and ν LP ( ≡ −ULP, 2 /kB T) is thenematic interaction parameter between a liquid crystal and apolymer. When ν LP > 0, it means the attractive interaction
between a liquid crystal and a polymer chain. The chiralinteraction parameter cLL ( ≡ −ULL, 1 /kB T > 0) corresponds
to the strength of the chirality between cholesteric liquid
crystal molecules and cLP ( ≡ −ULP, 1 /kB T) is that between
a liquid crystal and a polymer. The physico-chemical originof the nematic interaction ν LP comes from a straightening ofpolymer chains due to the anisotropic coupling with liquid
crystals. When a polymer chain is flexible, the strength of theorientational-dependent interaction is weak and we can takeν LP = 0. On the other hand, for a semiflexible polymer chain,
the value of ν LP can have a finite value. The larger ν LP value
implies that the nematic phase tends to be more stable in themixed state than in their pure phase.
In order to calculate the distortion free energy due to thespatial variation of the director in a cholesteric phase, we assume that the director is uniformly twisted along z axis withpitch p = 2π /q:n(r) = (cos qz, sin qz, 0),
and the tensor order parameter can be expressed as52


3
1
n
δ
Q(i)
(r
)
=
S
(r
)n
(r
)
−
i
α 1 β 1
αβ ,
αβ 1
2
2

(21)

(22)

where Si is the scalar orientational order parameter of the particle i( = L, P):
(23)
Si = P2 (n(r) · )fi (n(r) · )d.

174906-4

Akihiko Matsuyama

J. Chem. Phys. 139, 174906 (2013)

The scalar order parameters do not depend on r in our mean
field approximations.
Substituting Eqs. (21) and (22) into Eqs. (13) and (14),
we obtain the anisotropic free energy:
Fani = Fnem + Fdis ,(24)

where we have separated the anisotropic free energy into twoparts. One is the nematic free energy of Maier-Saupe type:50
 φi 
a 3 βFnem /V =fi (n(r) · ) ln 4πfi (n(r) · )drd
ni=L,P i1
− φL2 SL2 νLL − φL φP SL SP νLP ,(25)
2
and the other is the distortion free energy due to the spatial
variation of the director:
1a 3 βFdis /V = − φL2 SL2 gLL (Q) − φL φP SL SP gLP (Q),2
(26)
where we define Q ≡ qd0 ,1
gLL (Q) ≡ − νLL Q2 + cLL Q,
2

(27)

1
gLP (Q) ≡ − νLP Q2 + cLP Q.
2

(28)

and

B. Distribution functions in an equilibrium state

The orientational distribution function fL (n(r) · ) of theliquid crystals and fP (n(r) · ) of the polymer chains are determined by the anisotropic free energy (Eq. (24)) with respect
to these functions: (δFani /δfi ) = 0, under the normalization
condition:(29)
fi (n(r) · )d = 1.
We then obtain1
exp[nL (φL SL GLL (Q) + φP SP GLP (Q))P2 (x)]fL (x) =
ZL
(30)
and
fP (x) =1
exp[nP φP SP GLP (Q)P2 (x)],ZP

andJm [Q] ≡ 1[P2 (x)]m exp[nP φP SP GLP (Q)P2 (x)]dx,
0(35)
respectively, where m = 0, 1, 2. . . .
Substituting Eqs. (30) and (31) into (23), the scalar orientational order parameters SL and SP can be determined by
the two coupled- self-consistency equations:
SL = I1 [Q]/I0 [Q],(36)

SP = J1 [Q]/J0 [Q].

(37)

Using the distribution functions, Eqs. (30) and (31), theanisotropic free energy (Eq. (5) or Eq. (24)) for the cholestericphase is given bya 3 βFani /V =1 2 2
φ S GLL (Q) + φL φP SL SP GLP (Q)
2 L L
φL
φP
ln J0 [Q] −ln I0 [Q].
(38)
−
nP
nL

When Q = 0, Eq. (38) results in the nematic free energy.22, 25
The chemical potential of a liquid crystal molecule isgiven byβμL = β(∂F /∂NL )NP



1
1
1
φP + χ φP2= nL
ln φL +
−
nL
nL
nP
1
+ SL2 φL2 GLL (Q) + SL SP φL φP GLP (Q)21(39)− ln I0 [Q] ,nL
and that of a polymer:
βμP = β(∂F /∂NP )NL



1
1
1
φL + χ φL2= nP
ln φP +
−
nP
nP
nL
1
+ SL2 φL2 GLL (Q) + SL SP φL φP GLP (Q)21
ln J0 [Q] .
(40)
−
nP

(31)
C. Cholesteric pitch and twist elastic constantwhere we define
GLL (Q) ≡ νLL + gLL (Q)(32)

GLP (Q) ≡ νLP + gLP (Q).

(33)

and

The constants ZL and ZP are determined by the normalization
condition as ZL = 4π I0 [Q] and ZP = 4π J0 [Q], respectively.
The functions Im and Jm are defined as
 1
Im [Q] ≡[P2 (x)]m exp[nL (φL SL GLL (Q)0+φP SP GLP (Q))P2 (x)]dx(34)

The cholesteric pitch p∗ (= 2π d0 /Q∗ ) in an equilibrium
state can be derived by minimizing the anisotropic free energy
Fani (Eq. (38)) with respect to Q:
 
SP
1
−
1
−
2
φP
x
S
L
Q∗ = Q0,(41)
1 − 1 − 2 n SSPL φP
where we define Q0 ≡ cLL /ν LL , n ≡ ν LP /ν LL , and∗x ≡ cLP /cLL . We have Q = 0 for SL = 0. The equilibrium∗
value Q of the pitch depends on two orientational order parameters in the form SP /SL , hence it is temperature and concentration dependent. The cholesteric pitch p∗ can be given by

Akihiko Matsuyama

J. Chem. Phys. 139, 174906 (2013)

When x > n ( x < n ), the pitch p(∝1/Q) decreases (increases) with increasing φ P . The value of ( x − n )SP /SLcorresponds to the “microscopic twisting power” of the solute, depending on the nature of both solute and solvent
molecules.3 The value of the cholesteric pitch also strongly
depends on the chiral twisting power x between a polymer
and a liquid crystal. Using Eq. (41), the distortion free energy(Eq. (26)) yields


1
1
3
φL SL + n φP SP
a βFdis /V = νLL φL SL
2
2
×[(Q − Q∗ )2 − Q∗2 ].

(43)

When Q = Q∗ the total free energy has a minimum and thecholesteric phase with Q = Q∗ is always more stable than thenematic phase with Q = 0. There is no nematic-cholestericphase transitions in our model.
The term in the distortion free energy (Eq. (26)) proportional to Q2 can be identified as a bare twist elastic constant(K22 ) and then we obtain SPνLL 2 2φL φP . (44)SL φL + n(a/kB T )K22 =2
SLWhen φ P = 0, the twist elastic constant of the pure liquid
crystal molecule is given by52◦
(a/kB T )K22=

1
νLL SL2 .
2(45)

In Section III, we show some numerical results of thephysical properties of the cholesteric phase.
III. MIXTURES OF A CHOLESTERIC LIQUID CRYSTALAND A POLYMER

τ ≡ T /TCI = 4.55/(nL νLL ),(46)

where TCI shows the cholesteric-isotropic phase transition(CIT) temperature of a pure liquid crystal. The numerical parameters are given as a function of the temperature τ : cLL= Q0 ν LL , cLP = x cLL , and ν LP = n ν LL .

φ P=0.1

ε x=0.1

1.2
1.1

ε x=0.2

1.0

ε x=0.3

0.9
0.8
0.85

ε x=0.6

0.90

0.95 1.00
τ(=T/TCI)

1.05

1.10

FIG. 1. Cholesteric pitch p∗ /p0 (= Q0 /Q∗ ) plotted against temperature forvarious values of x for n = 0.3 and φ P = 0.1. The pitch strongly depends
on the chiral interaction parameter x .

A. Cholesteric pitch depending on concentrationand temperature

We first show the pitch of a cholesteric phase. Figure 1
shows the cholesteric pitch p∗ /p0 , Eq. (41), plotted against thetemperature τ at φ P = 0.1 for n = 0.3, nL = 2, and nP = 50.
The chiral interaction parameter x is changed. At the dottedline the CIT takes place and the pitch becomes infinity. Depending on the chiral interaction parameter x , we find three
types of temperature-dependent pitch. As shown in Eq. (42),the value of Q∗ decreases with increasing SP /SL for x < n at
a constant φ P . For a weak chiral coupling ( x = 0.1) between apolymer and a liquid crystal, the pitch increases with increasing temperature and disappears at the CIT. On the other hand,for a strong chiral coupling, or x > n ( x = 0.6), the pitch
decreases with increasing temperature. On decreasing temperature, the pitch becomes almost independent of temperature.
When x = n ( x = 0.3), the pitch is a constant.
Figure 2 shows the orientational order parameters (left)
and the ratio (SP /SL ) (right), plotted against temperature withn = 0.3 and φ P = 0.1. We have the first-order CIT atτ  1.04, where the orientational order parameters jump andincrease with decreasing temperature. The value of the ratio
SP /SL decreases with decreasing temperature and becomes almost a constant. As shown in Eq. (42), the temperature dependence of the pitch strongly depends on the sign of x − nand SP /SL .
Figure 3 shows the cholesteric pitch p∗ /p0 , Eq. (41),plotted against the polymer concentration at τ = 0.95 for
1.0
0.8
0.6

φ P=0.1

SP
SL

6.0
4.0

0.4

SP/SL

In our numerical calculations we have three interaction parameters related to cholesteric ordering: the chiral interaction parameter x , the nematic interaction n betweena polymer and a liquid crystal, and the pitch Q0 (≡cLL /ν LL = 2π d0 /p0 ) of the pure liquid crystal. We can estimate
as Q0  0.01 for the typical pitch p0 = 3000 Å and the length
d0 = 30 Å of a liquid crystal molecule. We here define thereduced temperature

1.3

p*/p0

p∗ /p0 = Q0 /Q∗ , where p0 is the pitch of the pure liquid crystal. When φ P = 0, the pitch of the pure liquid crystal is given
by p0 = 2π d0 /Q0 and a constant, or temperature independent.
By solving the coupled equations (36) and (37), using Q∗ , we
can obtain the values of two orientational order parameters SL
and SP as a function of temperature and concentration.
When φ P  1, we obtainSPQ∗  Q0 1 + 2( x − n ) φP · · · .
(42)SLOrder parameters

174906-5

2.0
0.2
0.0
0.85

SP/SL

0.90

0.95 1.00
τ(=T/TCI)

1.05

0.0
1.10

FIG. 2. Orientational order parameters (left) and the ratio (SP /SL ) (right),
plotted against temperature with n = 0.3 and φ P = 0.1. The order parameters decrease with increasing temperature and we have the first-order CIT atτ  1.04.

174906-6

Akihiko Matsuyama

J. Chem. Phys. 139, 174906 (2013)

2.0

1.0
Order parameters SP, SLε x=0.1

1.5

ε x=0.3

1.0

ε x=0.6

0.5
0.0

0.2

0.4
φP

0.6

0.8

0.8

τ=0.9
SP

2.0
1.5

SP/SL

0.6
0.4

1.0

SL

0.5

0.2
0.0
0.0

SP/SL

p*/p0

T/TCI=0.95

0.1

0.2

0.3
φP

0.4

0.0
0.5

FIG. 3. Cholesteric pitch p∗ /p0 ( = Q0 /Q∗ ) plotted against the polymer concentration for the various values of x with n = 0.3 and τ = 0.95.

FIG. 5. Orientational order parameters SP , SL , and the ratio SP /SL , plottedagainst the polymer concentration φ P for τ = 0.9 in Fig. 4.

n = 0.3, nL = 2, and nP = 50. At the dotted line the CIT
takes place. When x > n , the pitch decreases with increasing φ P , because of the strong chiral coupling between a polymer and a liquid crystal. When the chiral coupling parameteris weak, x < n , the helical pitch increases with φ P and unwinds at the CIT. These numerical results are consistent withthe experimental results.2, 8, 11–13 The pitch strongly depends
on the chiral interaction parameter x between a polymer anda liquid crystal.

the width of the two-phase region becomes wide. At the low
polymer concentrations, we have the stable cholesteric phase.
Figure 5 shows the orientational order parameters SP ofthe polymer and SL of the liquid crystal, plotted against thepolymer concentration at τ = 0.9 in Fig. 4. The orientational order parameters decrease with increasing the concentration of polymer chains and we have the first-order CIT at
φ P  0.19. The orientational order of the polymer is higher
than that of the liquid crystal, because of the coupling between the polymer and liquid crystal. For a weak coupling,or smaller values of n , the orientational order SP has smaller
values than that of SL and the binodal curve of the cholestericphase in Fig. 4 shifts to the lower concentrations.
Figure 6 shows the cholesteric pitch p∗ /p0 , Eq. (41), plotted against the volume fraction of polymers for various temperatures in Fig. 4. As shown in Eq. (42), the value of Q∗ decreases with increasing φ P for x < n , or the pitch increases
with increasing φ P . Figure 7 shows the twist elastic constant
K22 plotted against the volume fraction of polymers for various temperatures. The elastic constant decreases with the increasing polymer concentration. The value of K22 increases
with decreasing temperature at a fixed φ p . When (a/kB T)K22= 0.5, by taking kB T  1 kcal/mol and a  15 Å, we can estimate K22 ∼ 10−6 dyn, which is the correct order of magnitudeof the elastic constant.3 Our numerical results are consistent
with the experimental observations.

B. Phase separations

In this subsection we show the phase behaviors of mixtures of a polymer and a cholesteric liquid crystal. The equilibrium free energy is calculated by substituting Q∗ into
Eq. (38). The coexistence (binodal) curve can be calculated
from the standard common-tangent construction in the freeenergy (Eq. (1)) to find coexisting phases of different concentrations. We here take χ /ν LL = 0.1. The binodal curves
are also obtained by solving the two-phase coexistence conditions: the chemical potentials μL and μP of coexisting phases
have to be equal to each other.
Figure 4 shows the phase diagram on the temperatureconcentration plane for n = 0.3, x = 0.2, nL = 2, andnP = 10. Solid curves show the binodal line and the dottedline shows the CIT. The CIT temperature decreases with increasing polymer concentrations because of dilution. Belowτ < 1, we have the phase separation (Ch+I) between acholesteric (Ch) and an isotropic (I) phase. The CIT line ishidden inside the binodal lines. With decreasing temperature,

C. Polymer-induced cholesteric phases

When the nematic coupling between a polymer and aliquid crystal is strong, the phase behaviors are drastically

1.1
1.08
1.06

I

1.0

p*/p0

T/TCI

nP=10

Ch

0.9
0.0

1.04

Ch+I

τ=0.85

τ=0.9
τ=0.95

1.02
0.1

0.2

0.3
φP

0.4

0.5

FIG. 4. Phase diagram on the temperature-concentration plane for n = 0.3,
x = 0.2, nL = 2, and nP = 10. Solid curves show the binodal line and thedotted line shows the cholesteric-isotropic phase transition.

1.00
0.0

0.1

φP

0.2

0.3

FIG. 6. Cholesteric pitch p∗ /p0 , Eq. (41), plotted against the volume fractionof polymers for various temperatures in Fig. 4.

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J. Chem. Phys. 139, 174906 (2013)

1.10
nP=50
1.05
T/TCI

aK22/kBT

0.6
0.4

I

Ch

1.00
0.95

0.2
τ=0.95

0.0
0.0

τ=0.9

0.1

0.90

τ=0.85

0.2

φP

0.3

0.4

0.85
0.0

0.5

FIG. 7. Twist elastic constant K22 plotted against the volume fraction ofpolymers for various temperatures in Fig. 4.

changed. On increasing the molecular weight of a polymer chain, this anisotropic coupling becomes dominant. The
anisotropic coupling between a polymer and a liquid crystal has been widely studied in nematic and smectic A phasein polymer and liquid crystal mixtures. It has been recognized that smectic A and nematic phases can be induced
in a nematic liquid crystal mixture by strong mesogenic
interactions.26–29
Figure 8 shows the phase diagram on the temperatureconcentration plane for nP = 20. Solid curves show the binodal line and the dotted line shows the CIT. We have thephase separation (Ch+I). On increasing the molecular weightof the polymer, the anisotropic coupling between a polymer
and a liquid crystal becomes strong and the CIT line shifts
to higher temperatures and the binodal line of the cholestericphase shifts to higher concentrations, compared to nP = 10
(Fig. 4).
Figure 9 shows the phase diagram on the temperatureconcentration plane for nP = 50. Solid curves show the binodal line and the dotted line shows the CIT. We find that
the CIT temperature increases from τ = 1 with increasing
φ P and has a maximum as a function of φ P . The cholestericphase is stabilized by the anisotropic coupling between apolymer chain and a liquid crystal.53 The binodal lines ofthe phase separation (Ch+I) merge, or exhibit azeotropes, atwhich the two equilibrium isotropic and cholesteric phases
have the same composition, while the cholesteric-isotropic
phase separation (Ch+I) appears at high polymer concentrations. At the lower temperatures of the polymer inducedcholesteric phase, we have the phase separation between twocholesteric phases (Ch1 +Ch2 ) with different polymer concen-

Ch1+Ch2

0.2

0.4
φP

0.6

0.8

FIG. 9. Phase diagram on the temperature-concentration plane for n = 0.3,
x = 0.2, nL = 2, and nP = 50. Solid curves show the binodal line andthe dotted line shows the cholesteric-isotropic phase transition. The strong
anisotropic coupling between a polymer chain and a liquid crystal induces
the cholesteric phase at high temperatures.

trations. In this figure we have calculated with n = 0.3.
When the nematic interaction is weak, for example, n= 0.1, such induced-cholesteric phase does not appear and weonly have the cholesteric-isotropic phase separations, where
the CIT temperature decreases from τ = 1 with increasing
φ P , as shown in Fig. 4.
Figure 10 shows the cholesteric pitch p∗ /p0 , Eq. (41),plotted against the temperature for various values of the polymer concentration φ P in Fig. 9. As shown in Eq. (42), forx < n , the pitch increases with increasing φ P . Figure 11
shows the twist elastic constant K22 plotted against the volume
fraction φ P for various temperatures in Fig. 9. At low temperatures (τ < 1), the value of K22 decreases with the increasing
polymer concentration φ P . This is similar to Fig. 7. At hightemperatures (τ > 1), however, the K22 has a maximum as
a function of φ P due to the appearance of polymer-inducedcholesteric ordering.
In our theory, there are two important molecular parameters to understand cholesteric behaviors in the liquid crystal/polymer mixtures: one is the nematic interaction n andthe other is the chirality parameter x between a polymer anda liquid crystal. The chiral interaction x depends on the helical configuration between a polymer and a liquid crystal.
The phase diagrams including the CIT and binodal curves arealmost independent of the chiral interaction parameter x , because the contribution of the chirality in the free energy issmall (order of Q( 1)), although, the pitch strongly depends
on x . The larger values of x correspond to the smaller pitch,

1.05
1.2

nP=20
I

φ P=0.2

0.95 Ch

p*/p0

T/TCI

1.00

Ch+I

φ P=0.1

1.1

0.90
0.85
0.0

φ P=0.05

0.1

0.2

φP

0.3

0.4

0.5

FIG. 8. Phase diagram on the temperature-concentration plane for n = 0.3,
x = 0.2, nL = 2, and nP = 20. Solid curves show the binodal line and thedotted line shows the cholesteric-isotropic phase transition.

1.0
0.85

φ P=0.01

0.90

0.95 1.00
τ(=T/TCI)

1.05

1.10

FIG. 10. Cholesteric pitch p∗ /p0 , Eq. (41), plotted against the temperature
for various of the polymer concentration φ P in Fig. 9

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Akihiko Matsuyama

1.5

1.0

0.9

0.4

0.85

0.3

0.04

0.02

0.00
0.0

1.02

SP/SL

aK22/kBT

0.06

J. Chem. Phys. 139, 174906 (2013)0.15

τ=1.05

0.2

0.4
φP

0.6

0.5
0.85

0.8

FIG. 11. Twist elastic constant K22 plotted against the volume fraction ofpolymers for various temperatures in Fig. 9.

where the pitch decreases with increasing temperature andpolymer concentration.
IV. MIXTURES OF A NEMATIC LIQUID CRYSTALAND A CHIRAL DOPANT

Our theory presented in Sec. III can describe mixturesof a non-chiral nematic liquid crystal (i = L) and a chiraldopant (i = P). In this section we focus on the pitch of the
mixtures. It is well known that when a non-chiral nematic liquid crystal phase is doped with a low concentration of a chiral molecule, a chiral nematic phase is formed with a helicalpitch, which is inversely proportional to the concentration of
the chiral dopant.3 This chiral nematic phase is induced bya chiral coupling between the liquid crystal and dopant. In
Sec. III, we have considered the chiral coupling between two
constituent molecules and then our theory can give the underlying physics of the chiral coupling.
For non-chiral nematic liquid crystal molecules (i = L),
we can take ULL, 1 = 0 in Eq. (4), or cLL = 0 in Eq. (27). Thenthe cholesteric pitch (or the wave number Q of the cholestericpitch) is given as a function of the volume fraction φ P of thechiral dopant:
Q=

0.2

1.0

2 n αx SSPL φP ,1 − 1 − 2 n SSPL φP

HT P
φP + · · · ,
αx

(48)

SP
SL

(49)

where
HT P ≡ 2 n αx

0.90

0.95
T/TCI

1.00

1.05

FIG. 12. Ratio SP /SL of two orientational order parameters plotted against
the reduced-temperature for various values of n with φ P = 0.01.

the same with Fig. 4. We find that the temperature dependence of HTP has three types. When n = 0.3 and 0.4, or astrong coupling between a nematic liquid crystal and a chiraldopant, the value of SP /SL is larger than unity and increases
with increasing temperature because of the anisotropic coupling between a liquid crystal and a chiral dopant. The valueof HTP increases with increasing temperature. On the other
hand, for a weak coupling n = 0.1 and 0.15, the ratio of
two order parameters decreases with increasing temperature.
In this case, the polymer chains are weakly oriented due tothe weak anisotropic coupling with liquid crystal molecules.
The HTP decreases with increasing temperature. Whenn = 0.2, the value SP /SL is independent of temperature andthe helical twisting power becomes a constant. When SP /SL 1, α x = 1, and d0 = 0.004 μm, we can estimate the helical twisting power HTP = HTP /d0  100 (μm−1 ), which isconsistent with the magnitude of the helical twisting power innematic liquid crystals K15 induced by TADDOL dopants.56
The temperature dependence of the helical twisting power isdetermined by the nematic interaction parameter n between
a liquid crystal and a chiral dopant.
Using the twist elastic constant (Eq. (45)) of the pure nematic liquid crystal, we find that the helical twisting power
can be expressed as(47)

where α x (≡ cLP /ν LP ) shows the strength of the chirality of
dopants ( n α x = cLP /ν LL ).
When φ P  1, we find
Q/φP  HT P + HT P 1 −ε n=0.1

is called the helical twisting power of a chiral dopant.1, 3 Thelarger values of HTP correspond to the smaller pitch. For HTP> 0, the wave number Q of the twist pitch linearly increases
with increasing φ P at dilute solutions, which is consistent withthe experiments.1, 53–55 The temperature and concentration dependences of HTP are given through the ratio SP /SL .
Figure 12 shows the ratio SP /SL of two orientational orderparameters plotted against the reduced-temperature for various values of n with φ P = 0.01. The other parameters are

HT P =cLP
cLP SP
,◦◦aK22 /kB T
aK22 /kB T(50)

for SP  1. The helical twisting power is inversely propor◦ 57–60
and the sign of HTP
tional to the twist elastic constant K22
depends on the values of cLP and SP . Note that the positive
values of cLP correspond to an attractive chiral interaction between a nematic liquid crystal and a dopant. The sophisticatedcalculations of HTP have been studied by many authors.57–61
Following the simulation of Allen,58 the parameter cLP in ourmodel corresponds to the difference μ in the chemical potentials between mirror image forms of a dopant, where μchanges the sign as a function of twist angle between thesymmetry axes of two ellipsoids. Due to the surface chirality model developed by Ferrarini et al., the term cLP SP cancorrespond to the chirality order parameter, which describes
the coupling between the chiral surface of the molecule andits orientational ordering.60
As shown in Fig. 13, when Q/φ P is plotted against φ P ,using Eq. (48), the intercept corresponds to HTP and the slope

174906-9

Akihiko Matsuyama

J. Chem. Phys. 139, 174906 (2013)

Q/φ P

Λ
HTP
φP

0

FIG. 13. Wave number Q/φ P plotted against φ P for HTP > 0 (see Eq. (48)).
When HTP < 0, the slope becomes negative.

() shows the value



HT P
,
 = HT P 1 −
αx

(51)

yielding
αx =

HT P
.
1 − /HT P(52)

Using Eq. (49), we have
n =

HT P
HT P

.
2αx (SP /SL )
2αx

(53)

for SP /SL  1. Then if we experimentally observe HTP and ,
the strength of chirality cLP (or α x ) and the nematic interactionn can be determined from Eqs. (52) or (50) and (53). When  0, we have α x  HTP and the value of n is of the orderof unity.
V. SUMMARY

We have presented a mean field theory to describe phasebehaviors in mixtures of a polymer and a cholesteric liquid
crystal. Our main original contribution is to taking into account the chiral coupling between a polymer and a liquid
crystal. As the results, we obtain analytical expressions of thepitch of the cholesteric phase and the twist elastic constant as
a function of orientational order parameters of a polymer anda liquid crystal.
The pitch of the cholesteric phase depends on the chiral interaction parameter x and the nematic parameter n
and is given as a function of temperature and concentration
through the ratio of orientational order parameters in the form
SP /SL . When x > n ( x < n ), the pitch decreases (increases)
with increasing the polymer concentration and temperature.
We also find isotropic-cholesteric and cholesteric-cholestericphase separations on the temperature-concentration plane. We
demonstrate that a strong coupling between a polymer and aliquid crystal can stabilize cholesteric phases.
Our theory can apply to mixtures of a non-chiral nematic
liquid crystal and a chiral dopant. We have derived the pitch
as a function of the concentration of chiral dopants and theorientational order parameters, and the chiral interaction parameters. We have shown that these chiral interaction parameters can be determined by experiments. We find that the HTP
is given as a function of a ratio of orientational order parameters in the form SP /SL , which depends on temperature andconcentration. For a strong (weak) nematic coupling between

a nematic liquid crystal and a chiral molecule, the HTP increases (decreases) with increasing temperature. Our theory
is consistent with some experimental observations.
In this paper we have considered the equilibrium properties of cholesteric phases in mixtures of a liquid crystal
and a polymer. The kinetics of the phase ordering process
is also important. Luca and Rey have used a model basedon the Landau-de Gennes theory to understand the structureformation process of the twisted polywood architecture ubiquitously found in biological fibrous composites.32, 33 They
found that the phase ordering process is slowed by an increasein chirality, linked to a biaxiality. We hope that our theory will
be applied to understand the kinetics of phase separations including cholesteric phases in mixtures of a liquid crystal anda polymer.
We here do not take into account biaxiality42, 62 of acholesteric phase and also neglect excluded volume interactions, which are important for long rodlike molecules.51 This
entropic contribution affects the temperature dependence of acholesteric pitch.43 The theory can be extended to nematicand cholesteric liquid crystal mixtures, binary mixtures ofcholesteric liquid crystals, cholesteric liquid crystalline polymers, and lyotropic cholesteric phases.
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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