Phase diagrams of mixtures of a polymer and a cholesteric liquid crystal under anexternal fieldAkihiko Matsuyama
Citation: The Journal of Chemical Physics 141, 184903 (2014); doi: 10.1063/1.4901085
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THE JOURNAL OF CHEMICAL PHYSICS 141, 184903 (2014)

Phase diagrams of mixtures of a polymer and a cholesteric liquid crystal
under an external fieldAkihiko 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 15 July 2014; accepted 24 October 2014; published online 12 November 2014)
We present a mean field theory to describe phase behaviors in mixtures of a polymer and a cholestericliquid crystal under an external magnetic or electric field. Taking into account a chiral coupling between a polymer and a liquid crystal under the external field, we examine twist-untwist phase transitions and phase separations in the mixtures. It is found that a cholesteric-nematic phase transition canbe induced by not only the external field but also concentration and temperature. Depending on thestrength of the external field, we predict cholesteric-paranematic (Ch+pN), nematic-paranematic(N+pN), cholesteric-nematic (Ch+N) phase separations, etc., on the temperature-concentration
plane. We also discuss mixtures of a non-chiral nematic liquid crystal and a chiral dopant. © 2014
AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4901085]
I. INTRODUCTION

The behaviors of liquid crystals in the presence of an
external field are of important for many technological applications as well as fundamental physical researches.1 The effects of an external field have been studied on the nematicisotropic phase transition (NIT) of nematic liquid crystals,2, 3
the cholesteric-nematic phase transitions (CNT) of cholestericliquid crystals,4–10 liquid crystalline elastomers,11–13 and thephase diagrams of mixtures of polymers and nematic liquid
crystals.14–16
The influence of an external magnetic or electric field ona cholesteric liquid crystal has been studied theoretically17–27
and experimentally.28–32 This problem in an external field was
first studied by de Gennes6 and Meyer.7–9 Applying the external field perpendicular to a helical axis of the cholestericphase, untwisting of the helix takes place with positive dielectric or diamagnetic anisotropy. We here assume that the
thickness of the sample is sufficiently large and the boundary
conditions can be neglected. On increasing the external field
H, the pitch p increases and diverges at a critical value Hc ofthe external field. When H > Hc , we have a nematic phase,oriented uniformly along the external field. This is known asa field-induced twist-untwist transition (or CNT),10 which has
been used in liquid crystal displays.
Most experiments of the field-induced CNT have been
carried out, not on pure cholesteric liquid crystals, but on mixtures of a nematic liquid crystal and a chiral dopant.7, 8, 31 It
is well known that the critical field Hc is a linear functionof the concentration of dopants in the dilute regime and thepitch of a helix is inversely proportional to the concentration.1
There are many studies of the CNT based on the Frank’s elastic theory,1, 17–26 however, it cannot describe the concentrationa) Electronic mail: matuyama@bio.kyutech.ac.jp. URL: http://iona.bio.

kyutech.ac.jp/~aki/.

0021-9606/2014/141(18)/184903/9/$30.00

dependences. To describe the concentration dependences, weneed to consider binary mixtures of a liquid crystal and another molecule. Recently, developing the molecular theoryof the cholesteric phase presented by Lin-Liu et al.,33, 34 wehave presented the molecular theory to describe phase behaviors of binary mixtures of a liquid crystal and a polymer, by taking into account of the chiral coupling between
the components.35 We have obtained the expression of thepitch of a cholesteric phase and the twist elastic constant as afunction of orientational order parameters and concentrations
and found various phase separations between a cholestericand an isotropic phases (Ch+I) and two cholesteric phases(Ch1 +Ch2 ).
In this paper, we develop our previous theory35 to describe cholesteric phases in mixtures of a cholesteric liquid
crystal and a polymer chain in the presence of an external
magnetic or electric field. We derive the free energy of the
mixtures under the external field and calculate the pitch andthe critical field as a function of orientational order parametersand concentrations. We predict the field-induced CNT on thetemperature-concentration plane and novel phase behaviors,such as a cholesteric-paranematic (Ch+pN) and cholestericnematic (Ch+N) phase separations, etc., depending on thestrength of the external field. Our theory can also apply to
mixtures of a non-chiral nematic liquid crystal and a chiraldopant, where the cholesteric phase is induced by the chiral coupling between the liquid crystal and dopant under anexternal field. Our results are consistent with experimental
observations.
In Sec. II, we extend a mean field theory to describe binary mixtures of a cholesteric liquid crystal and a polymer
chain in the presence of an external field. In Sec. III, we show
some numerical results of pitch, the strength of a critical field,
and phase diagrams on the temperature-concentration plane
for the mixtures. In the Appendix, we discuss mixtures of anematic liquid crystal and a chiral dopant.

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184903-2

Akihiko Matsuyama

J. Chem. Phys. 141, 184903 (2014)

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 liquid crystalline ordering isforbidden in the constituent pure polymers, such as a flexible
polymer chain. To describe cholesteric phases of the mixtures,
we develop a mean field model proposed by Lin-Liu et al.33, 34
We here take into account an anisotropic coupling between
polymers and liquid crystals.
Let NP be 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 volumeof the liquid crystal and that of the polymer chain is given
by vL = (π/4)D 2 L and vP = a 3 nP , respectively, where a3
is the volume of a polymer segment. Let φL = vL ρL andφP = vP ρP be the volume fraction of the liquid crystal andthe polymer, respectively, where ρ i is the number density
ρi = Ni /V of the molecule i(= L, P): φ L + φ P = 1. Using
the axial ratio nL = L/D of the liquid crystal, the volume per aliquid crystal molecule is given by vL = a 3 nL where we here
assume a3 = (π /4)D3 for simplicity.
The free energy consists of the following three terms:
F = Fmix + Fani + Fext .

(1)

The first term in Eq. (1) is the free energy of an isotropic
mixing of a polymer and a liquid crystal molecule and is given
by Flory-Huggins theory for polymer solutions36
φ
φa βFmix /V = L ln φL + P ln φP + χ φL φP ,nLnP
3

Uij (r1 , 1 ; r2 , 2 ) = Uij,1 (r12 )(1 × 2 · r̂12 )P1 (1 · 2 )+ Uij,2 (r12 )P2 (1 · 2 ),(4)

where we have truncated by P2 (x). The potential Uij, 1 shows
the chiral interaction between two particle i and j. The term
1 × 2 · r̂12 represents scalars coupling between orientational and spatial variables, where r̂12 = (r2 − r1 )/|r12 |. The
potential Uij, 2 shows the intermolecular potential that accounts for the formation of a nematic phase. We here assume
that the potentials are short range interactions which havebeen used in Maier-Saupe38 and Onsager models.39 We employ a simple square wall interaction potential with a shortrange d0 , which is the order of the particle size. Following
the symmetry consideration for cholesteric phases, we requirethat the first term in Eq. (4) is odd in 1 · 2 and the second
term be even.37
The last term in Eq. (1) is the magnetic (or electric) freeenergy relevant to orientational order. When the external magnetic field H is applied to the molecule i(= L, P) having adiamagnetic anisotropy χ i , the external free energy is givenby1
Fext = −ni ρi(5)χi Hα Q(i)αβ (r)Hβ dr,
i=L.P

(2)

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 the constituent
molecules 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 the 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 is given by
ρi fi (n(r) · ) ln 4πfi (n(r) · )drd
βFani /V =i=L,Pwhere Q(i)αβ (r) is the second rank order parameter tensor of themolecule i1


3
1
,
(6)
n
δ
Q(i)
(r)
=
S
(r)n
(r)
−
i
β
αβ
2 α
2 αβ
and Si is the scalar orientational order parameter of the particle i(= L, P)
(7)Si = P2 (n(r) · )fi (n(r) · )d.
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 with apitch p = 2π /qn(r) = (cos θ, sin θ, 0).

(3)where dR ≡ dr1 dr2 d1 2 . The first term in Eq. (3) showsthe entropy changes due to an orientational ordering and Uijis the orientation-dependent intermolecular potential between(8)

The scalar order parameters do not depend on position r andθ is a function of z in our mean field approximations. We here
set the magnetic field H is taken along y axis
H = (0, H, 0).

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

×βUij (r1 , 1 ; r2 , 2 )dR,

two particles i and j (i, j = L, P). We here take UPP = 0 because we consider non-nematic polymer chains for the constituent pure polymers. The lowest-order contributions to theinteraction potential for the cholesteric phase are given by ina series of the Legendre polynomials33, 35(9)When χ i > 0, the molecules are oriented along the external
field.
Substituting Eqs. (6) and (9) into Eqs. (3)–(5), we obtain the free energy of the cholesteric phase (refer to previous
paper35 for details)
Fch ≡ Fani + Fext = Fnem + Fdis ,(10)This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
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184903-3

Akihiko Matsuyama

J. Chem. Phys. 141, 184903 (2014)

where we have separated the free energy into two parts forconvenience. One is the nematic free energy of Maier-Saupe
type38 φ ia 3 βFnem /V =fi (n(r) · ) ln 4πfi (n(r) · )dni=L,P i1
− νLL φL SL (φL SL + 2n φP SP ),(11)2
where the interaction parameter ν LL (≡ −βULL, 2 > 0) corresponds to the orientational-dependent (Maier-Saupe)
interaction parameter between liquid crystal molecules38 and n ≡ν LP /ν LL : the parameter ν LP (≡ −βULP, 2 ) is theorientational-dependent interaction parameter between a liquid crystal and a polymer. When ν LP > 0, it means the attractive interaction between a liquid crystal and a polymer chain.
The second term of Eq. (10) shows the distortion energy
due to the spatial variation of the director in the presence ofan external field1
a 3 βFdis /V = − νLL φL SL (φL SL + 2n φp SP )g(Q) (12)2where we define Q ≡ qd0 . The function g(Q) contains thespatial variation of the twist angle and is given as  1/Q  2
∂θ
Q
∗ ∂θ
− 4π Qg(Q) ≡ − 2
8π 0
∂ z̃
∂ z̃
 2 
2π
1−
sin2 θ −
d z̃,
(13)
ξ
3
Q ∗ ≡ Q0φL SL + 2x φP SP,
φL SL + 2n φP SP

(14)

and
ξ2 ≡

νLL φL SL (φL SL + 2n φP SP )
,
2 φL SL h2L + φP SP h2P(15)

where we define z̃ ≡ z/(2π d0 ), Q0 ≡ cLL /ν LL , and  x≡ cLP /cLL . The chiral interaction parameter cLL (≡ −βULL, 1> 0) corresponds to the chirality between cholesteric liquid
crystal molecules and cLP (≡ −βULP, 1 ) is that between a liquid crystal and a polymer. The larger values of cij shows thestronger chiral coupling between molecules. The value of Q∗
has the same form with the cholesteric pitch (p∗ = 2π d0 /Q∗ )of the mixtures in the absence of an external field and Q0
shows the pitch of the pure cholesteric liquid crystals in theabsence of an external field.35 The value of ξ is the magnetic coherence length.1 The external field is given throughthe parameters h2L ≡ 3β χL H 2 and h2P ≡ 3β χP H 2 forχ i > 0.
B. Cholesteric pitch

The cholesteric pitch p(= 2π d0 /Q) in an equilibrium state
can be derived by minimizing the distortion free energy Fdis
(Eq. (12)). The function θ (z) should satisfy the condition forminimum Fdis and the corresponding Euler equation is givenby 2  2ξ∂θ+ sin2 θ = c,
(16)
2π
∂ z̃

where c is an arbitrary constant. The pitch p of the cholestericphase satisfying the condition of a minimum in Fdis is
 
 p
 1/Q
∂ z̃
dθ,
(17)
p=
dz =
2π d0
∂θ
0
0
and using Eq. (16) we obtain
 
1
2
=
ξ K(c),
Q
π(18)

where K(c) is the complete elliptic integral of the first kind
 π/2dθK(c) =.
(19)0c − sin2 θ
From the behavior of K(c), the value of 1/Q clearly diverges
as c → 1.
Substituting Eqs. (16) and (19) into (13), we obtain




1
1
2
∗
g(Q) = − 2− c − Q −Q +E(c) , (20)2ξ
3πξwhere E(c) is the complete elliptic integral of the second kind π/2E(c) =
c − sin2 θdθ.
(21)
0

The constant c can be determined by dg(Q)/dc = 0 and wethen obtainπE(c) = ξ Q∗
(22)
2
and
p
Q
Q
4
= 0 = 2 K(c)E(c) ∗0
p0
Q
π
Q


φ S + 2n φP SP
4= 2 K(c)E(c) L L,(23)π
φL SL + 2x φP SP
where p0 = 2π d0 /Q0 is the pitch of a pure liquid crystal in theabsence of the external field. When φ P = 0, Eq. (23) resultsin the pitch of the pure liquid crystal derived by de Gennes.1
Substituting Eq. (22) into (20), the function g(Q) for theminimum of the distortion free energy is given by
11g(Q) = 2 c −
,
(24)
2ξ
3
where the constant c is determined by Eq. (22).
The value of second elliptic integral (21) is larger than 1
and increases with increasing c from c = 1. When (π /2)ξ Q∗> 1, the value of c is determined by Eq. (22) and the pitch Q
of the cholesteric phase is given by Eq. (18). When (π /2)ξ Q∗< 1, we have Q = 0 and c = 1 from Eq. (16). The cholestericphase is defined by Q > 0, SL = 0, and SP = 0, the nematic
phase is given by Q = 0, SL = 0, and SP = 0, and the isotropic
phase is defined by Q = SL = SP = 0. The orientational orderparameters are determined by minimizing the free energy asshown in Subsection II C.
C. Orientational distribution functionsin an equilibrium state

The orientational distribution function fL (n(r) · ) of theliquid crystals and fP (n(r) · ) of the polymer chains areThis article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
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184903-4

Akihiko Matsuyama

J. Chem. Phys. 141, 184903 (2014)

determined by the cholesteric free energy (Eq. (10)) with respect to these functions: (δFch /δfi ) = 0, under the normalization conditionfi (n(r) · )d = 1.
(25)
We then obtain the distribution function of the liquid crystal
moleculesfL (x) =1
exp[nL νLL (φL SL + n φP SP )ZL
× (1 + gL (Q))P2 (x)],Using Eqs. (16), (18), and (22) we obtain1
1gL (Q) = 2 c −
3
2ξ1




1
1 ξ
+
− Q −Q∗1 +
E(c)
ξ2
ξ12 π
for Eq. (27) andgLP (Q) =(26)

where we define 
 1/Q  2
∂θ
Q
∗ ∂θ
gL (Q) ≡ − 2− 4π Q1
8π 0
∂ z̃
∂ z̃

 2 
2π1
sin2 θ −
d z̃,
(27)
−
ξ1
3
Q∗1 ≡ Q0φL SL + x φP SP,
φL SL + n φP SP

νLL (φL SL + n φP SP )
.
2h2L

(29)

a 3 βFch /V =−1
exp[nP νLP φL SL (1 + gLP (Q))P2 (x)], (30)
ZP

where we define 
 1/Q  2
∂θ
∂θ
Q
−
4π
α
x
8π 2 0∂ z̃∂ z̃ 2 
2π1
sin2 θ −
d z̃,
(31)
−
ξ2
3

gLP (Q) ≡ −
ξ22 ≡ νLP φL SL / 2h2P .

(32)

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 νLL (φL SL + n φP SP )0

× (1 + gL (Q))P2 (x)]dx,

Jm [Q] ≡respectively, where m = 0, 1, 2, . . . .

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

(38)

φP
φ
ln J0 [Q] − L ln I0 [Q].
nP
nL

(39)

The total free energy F of our system is given by the sum of
Eqs. (2) and (39).
The chemical potential of the liquid crystal molecule isgiven byβμL = β(∂F /∂NL )N
P



1
1
1
= nL
ln φL +−
φP + χ φP2
nL
nL
nP

1
+ φL SL (φL SL + 2n φP SP ) 1 + g(Q)21−
ln I0 [Q] ,
nL

(40)

and that of the polymer
βμP = β(∂F /∂NP )N
L



1
1
1
φ + χ φL2
= nP
ln φP +
−
nP
nP
nL L
1
+ φL SL (φL SL + 2n φP SP )(1 + g(Q))
21
ln J0 [Q] .
(41)
−
nP

(34)

In Sec. III, we show some numerical results of the physical properties of the cholesteric phase in the presence of theexternal field.

[P2 (x)]m exp[nP νLP φL SL

× 1 + gLP (Q) P2 (x)]dx,

(37)

(33)

 1
0

SL = I1 [Q]/I0 [Q],1
νLL φL SL φL SL + 2n φP SP (1 + g(Q))
2

The distribution function of the polymer chains is given

α x ≡ cLP /ν LP , and(36)

Using the distribution functions, Eqs. (26) and (30), thefree energy (Eq. (10)) of the cholesteric phase is given by

byfP (x) =



1
1
c
−
3
2ξ22




1
1 ξ
+
− Q −αx +
E(c)
ξ2
ξ22 π

for Eq. (31).
Substituting Eqs. (26) and (30) into Eq. (7), the scalarorientational order parameters SL and SP can be determinedby the two coupled-self-consistency equations

(28)

and
ξ12 ≡

(35)

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184903-5

Akihiko Matsuyama

J. Chem. Phys. 141, 184903 (2014)

III. MIXTURES OF A CHOLESTERIC LIQUID CRYSTALAND A POLYMER

1.0

In our numerical calculation, we have three interaction
parameters related to cholesteric ordering: the chiral interaction parameter  x = cLP /cLL , the nematic interaction  n= ν LP /ν LL between a polymer and a liquid crystal, andQ0 (≡ cLL /ν LL = 2π d0 /p0 ) is the pitch 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.
The cholesteric phase of a pure liquid crystals in the absence of an external field appears at

0.8

nL ULL,2
 4.55,nL νLL =kB TCIτ ≡ T /TCI = 4.55/(nL νLL ).

(43)

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 . The
estimations of the interaction parameters  n and  x have been
discussed in our previous paper35 and are of the order of
0.1–1, resulting the twist elastic constant K22 ∼ 10−6 dyn ofa typical value.
A. Twist-untwist transitions and a critical field

In this subsection, we first show the twist-untwist transitions in the mixtures. For numerical calculations, we here takenL = 2,  n = 0.3, and hP = 0 (note that when hP = 0 we have
1/ξ22 = 0).
Figure 1 shows order parameters SL , SP , and the pitch
Q/Q0 (∝ p−1 ) plotted against the temperature T/TCI at thepolymer concentration φ P = 0.001 for  x = 0.3, nP = 10,
and hL = 0.014. The pitch Q is normalized by Q0 of thepure liquid crystal in the absence of the external field. At
low temperatures, we have the cholesteric phase with SL > 0,
SP > 0, and Q > 0. As increasing temperature, the value of Q
decreases and becomes zero, where the cholesteric (Ch) phasechanges to the nematic (N) phase with Q = 0. We find thehL=0.014

Order ParametersSP

φ P =0.001

0.6
SL

0.4
Q/Q0

0.2
0.0
0.85

SL

0.95

1.00

Order Parameters

SL
0.4
Q/Q0
0.2
SP
0.0
0.0

0.1

0.2

φP

SL
0.3

0.4

0.5

1.05

FIG. 2. Order parameters SL , SP , and the pitch Q/Q0 (∝ p−1 ) plotted against
the concentration φ P at the temperature T/TCI = 0.84 for  x = 0.3, nP = 10,
and hL = 0.014.

twist-untwist CNT, which depends only on the pitch while
the orientational order parameters continuously change. Further increasing temperature, orientational order parameters
jump to nonzero values and the system shows a weak nematic(or paranematic: pN) phase. We find the nematic-paranematic
phase transition (NpNT). The order parameter SP is larger
than SL because of the anisotropic coupling  n between thepolymer and liquid crystal. The NpNT is the first-order phasetransition, while the CNT is the second-order one.
Figure 2 shows order parameters SL , SP , and the pitch
Q/Q0 (∝ p−1 ) plotted against the concentration φ P at the temperature T/TCI = 0.84 for  x = 0.3, nP = 10, and hL = 0.014.
On increasing the polymer concentration, the value of Q decreases and we have the CNT and NpNT. As shown in Figs. 1
and 2, we find that the CNT and NpNT can be induced by temperature and concentration. On increasing the external fieldhL , the CNT shifts to lower temperatures and lower concentrations. When hL = 0, we only have the CIT.35
The critical field strength (hcL ), where the pitch p of atwist in the cholesteric tends to infinity, or Q = 0, is given by
E(1) = 1 in Eq. (22)
hcL =

π ∗ 1
Q
ν (φ S + 2n φP SP ).
2
2 LL L L(44)

When hL < hcL we have the cholesteric phase, while hL> hcL the untwisting of the cholesteric twist takes place andthe nematic phase appears. The term in the distortion free energy (Eq. (12)) proportional to g(Q) can be identified as a baretwist elastic constant (K22 ) and then we obtainνK̃22 = LL SL φL φL SL + 2n φP SP ,(45)
2
where K̃22 ≡ a(K22 /kB T ). Using the twist elastic constant,
we obtain the critical value (44) of the external field for the
mixtures

SP

0.90

SP
0.6

(42)

where TCI shows the cholesteric-isotropic phase transition(CIT) temperature of a pure liquid crystal in the absence ofan external field. We then define the reduced temperature0.8

hL=0.014
T/TCI=0.84

1.10

T/ TCI

FIG. 1. Order parameters SL , SP , and the pitch Q/Q0 (∝ p−1 ) plotted against
the temperature T/TCI at the polymer concentrations φ P = 0.001 for  x = 0.3,
nP = 10, and hL = 0.014.

hcL =π ∗ K̃22
Q
.
2
φL SL

(46)

The orientational order parameters can be calculated by
Eqs. (37) and (38) with Q = 0, c = 1, and ξ1,c ≡ ξ1 (hcL ). When

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Akihiko Matsuyama

J. Chem. Phys. 141, 184903 (2014)for  x = 0.2. On increasing the strength of the external field,
the pitch p diverges to infinity (Q → 0) and the CNT takes
place due to the external field. The orientational order parameters slightly increase with increasing hL . As shown in
Figs. 1, 2, and 4, our theory demonstrates that the CNT canbe induced by not only external field but also temperature andconcentration.

0.020
T/TCI=0.86

ε x=1.0

ε n=0.3

hc

0.015
0.6
0.4
0.2

0.010

0.005

0.000
0.0

B. Phase separations under an external field
0.1

0.2

φ

0.3

0.4

0.5

P

FIG. 3. Critical field hcL as a function of φ P for various values of the chiralinteraction  x at T/TCI = 0.86 for  n = 0.3 and nP = 10.

φP

1, Eq. (14) can be expanded asSP∗
Q  Q0 1 + 2(x − n ) φP · · · .
SL

(47)

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.1 Then the strength of the critical field depend on
the chiral interaction parameter  x .
Figure 3 shows the critical field hcL as a function of φ P forvarious values of the chiral interaction  x at T/TCI = 0.86 for n = 0.3 and nP = 10. When hL < hcL we have the cholestericphase, while hL > hcL the untwisting of a cholesteric twist
takes place and the nematic phase appears. When  x = 0.2,
the value of the critical field decreases with increasing φ P andjumps to zero at φ P  0.26. When φ P > 0.26, the system always shows a nematic or paranematic phase in the presence ofan external field. When the chiral interaction between a polymer and a liquid crystal is strong ( x = 1), the critical field hcL
has a maximum as a function of φ P .
Figure 4 shows the order parameters SL , SP , and the pitch
wavenumber Q/Q0 (∝ p−1 ) plotted against the external fieldhL at the polymer concentrations φ P = 0.01, T/TCI = 0.95

In this subsection, we show some numerical results ofthe phase diagrams on the temperature-concentration plane
for the mixtures of a polymer and a cholesteric liquid crystal under an external field. The equilibrium free energy is
calculated by substituting Q into Eq. (39). The coexistence(binodal) curve can be calculated by the standard commontangent construction in the free energy (Eq. (1)) to find coexisting phases of different concentrations. We here take χ /ν LL= 0.1. The binodal curves can also be obtained by solving thetwo-phase coexistence conditions: the chemical potentials μLand μP of coexisting phases have to be equal to each other.40
The phase diagrams in the absence of an external field
have been discussed in our previous paper,35 where we findthe phase separation (Ch+I) between a cholesteric and an
isotropic phase. Under an external field, the CNT takes place,depending on the temperature and concentration, as shownin Figs. 1 and 2, and then the phase diagrams are drasticallychanged.
Figure 5 shows the phase diagram on the temperatureconcentration plane for  x = 0.2 and nP = 10 in the presenceof a weak external field hL = 0.01. The solid curves are the
binodal line, the dotted line shows the NpNT, and the dashedline is the CNT, which corresponds to the critical field in
Fig. 3. The NpNT temperature of the pure liquid crystal(φ P = 0) shifts to higher temperatures because of the external field. When hL = 0, we have the CIT at T/TCI = 1 and the(Ch+I) phase separation appears below T/TCI = 1.35 For the
weak external field hL = 0.01, the nematic phase appears on
the narrow region between the Ch and pN phases. The CNT

1.05

1.2

N + pN

T/TCI=0.95
φ P=0.001

Q/Q0

0.8

SP

0.6

N

pN

Ch + pN

0.95
SL

0.4

Ch

0.90
0.00

0.2
0.0
0.000

1.00
T/ TCI

Order Parameters

1.0

hL=0.01
nP=10

Ch+N+pN

0.005

0.010

0.015

0.10

φP

0.20

0.30

0.020

hL
FIG. 4. Order parameters SL , SP , and the pitch Q/Q0 (∝ p−1 ) plotted against
the external field hL at the polymer concentrations φ P = 0.001, T/TCI = 0.95
for  x = 0.2.

FIG. 5. Phase diagram on the temperature-concentration plane for x = 0.2 and nP = 10 in the presence of a weak external field hL = 0.01.
The cholesteric, nematic, and paranematic states are denoted by Ch, N, andpN, respectively. The solid lines represent binodal lines, dotted lines show
the first-order phase transitions, and dashed lines correspond to continuous
phase transitions.

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Akihiko Matsuyama

J. Chem. Phys. 141, 184903 (2014)

1.15

1.05

hL=0.014

hL=0.014
nP=10

1.00

1.10
1.05

N + pN

0.90

N

T/ TCI

T/ TCI

pN
0.95

pN+N

pN
N

1.00
0.95
N1+N2

Ch+N+pN
0.85

0.2

φP

Ch+N
N+pN

0.85 Ch
0.0

Ch
0.1

Ch+N1+N2

0.90

Ch + pN

0.80
0.0

nP=50

pN

0.3

0.4

0.2

0.4
φ

0.6

0.8

P

0.5

FIG. 6. Phase diagram on the temperature-concentration plane for  x = 0.2
and nP = 10 in the presence of an external field hL = 0.014.

and NpNT temperature decrease with increasing the polymer
concentration φ P . We find the phase separation (N+pN) between a nematic and a paranematic phase at 1.025<τ <1.03
and the cholesteric-paranematic (Ch+pN) phase separationsat τ < 1.025. The N+pN coexistence is known in mixturesof a nematic liquid crystal and a polymer under external
fields.14 At low polymer concentrations, the stable Ch andN phases appear. We also have the three phase coexistence(Ch+N+pN) between Ch, N, and pN phases at T/TCI  1.025.
As shown in Figs. 1 and 2, the orientational order parametersin the pN phase have small values.
Figure 6 shows the phase diagram on the temperatureconcentration plane for  x = 0.2 and nP = 10 with hL= 0.014. On increasing the strength (hL ) of the external field,
the CNT temperature shifts to lower temperatures and the nematic phase appears on a broad region between the pN and
Ch phases. We have the N+pN and Ch+N phase separations.
At τ  0.87, we find three-phase coexistence (Ch+N+pN)
between Ch, N, and pN phases. As shown in Figs. 1 and 2,
the orientational order parameters jump at the NpNT and thepitch become infinity at the CNT.
Figure 7 shows the phase diagram on the temperatureconcentration plane for  x = 0.2 and nP = 20 with hL = 0.014.

FIG. 8. Phase diagram on the temperature-concentration plane for  x = 0.2
and nP = 50 in the presence of an external field hL = 0.014.

On increasing the molecular weight of the polymer nP , theanisotropic coupling between a polymer and a liquid crystal becomes strong and the NpNT line shifts to higher temperatures and we have broad biphasic regions, compared tonP = 10 (see Fig. 6).
Further increasing the molecular weight of the polymer,
the phase behavior is drastically changed. Figure 8 shows thephase diagram on the temperature-concentration plane for  x= 0.2 and nP = 50 with hL = 0.014. We find that the NpNT
temperature increases from τ = 1.03 with increasing φ P and
the pN+N phase separation appears. The NpNT line terminates at a critical point (closed circle), where the orientational order parameters are continuously changed (see Fig. 9).
The nematic phase is stabilized by the anisotropic coupling
between a polymer and a liquid crystal in the presence ofthe external field. At high concentrations, the NpNT linewith a critical point decreases with increasing φ P and the(pN+N) phase separations appears. At the lower temperaturesof the N phase, we have the phase separation between two nematic phases (N1 +N2 ) with different polymer concentrations
and Ch+N phase separation. We also find the triple point at
τ  0.89, where three phases (Ch+N1 +N2 ) coexist. On decreasing the strength of the external field hL , the CNT line
moves to upward and merges into the NpNT line (see Fig. 5).
When hL = 0, the nematic phase and critical points disappear
and we only have a cholesteric and an isotropic phases, as

1.05
hL=0.014
nP=20

pN

0.95 N

0.3

ε x=0.3

N + pNS

T/ TCI

1.00

0.4

0.90

0.2

Ch + N + pN

Ch
0.1

0.2

0.3
φP

1.09
1.10
T/TCI=1.11

0.1

Ch + pN

0.85
0.80
0.0

1.07
1.08

0.4

0.5

0.6

FIG. 7. Phase diagram on the temperature-concentration plane for  x = 0.2
and nP = 20 in the presence of an external field hL = 0.014.

0.0
0.0

0.1

0.2

0.3

φP

0.4

0.5

0.6

FIG. 9. The average orientational order parameter S(= SP φ P + SL φ L ) plottedagainst φ P for various temperatures near the critical points in Fig. 8. The
closed circles show critical points.

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Akihiko Matsuyama

J. Chem. Phys. 141, 184903 (2014)1.05
hL=0.014
nP=20pN

T/ TCI1.00

ε x=0.6

N+pN

0.95

N
Ch+N+pN

0.90

Ch+pN
Ch

0.85
0.0

0.1

0.2

0.3

φP

0.4

0.5

0.6

FIG. 10. Phase diagram on the temperature-concentration plane for  x = 0.6
and nP = 20 in the presence of an external field hL = 0.014. The open circle
shows critical end point.

shown in Fig. 9 of the previous paper.35 When the anisotropic
coupling  n between a polymer and a liquid crystal is zero,
the NpNT temperature decreases with increasing the polymer
concentration (see Figs. 5–7) and the critical points disappear.
Figure 9 shows the average orientational order parameter
S(= SP φ P + SL φ L ) plotted against φ P for various temperatures near the critical points in Fig. 8. Below τ < 1.1, the order parameter jumps at the two NpNT concentrations and has
a maximum between them. At τ = 1.1, we have two criticalpoints, where the order parameter continuously changes as afunction of φ P and temperature.
When the chiral coupling ( x ) between a polymer and aliquid crystal is strong, a cholesteric phase becomes stable
at higher temperatures. Figure 10 shows the phase diagramon the temperature-concentration plane for  x = 0.6 and nP= 20 with hL = 0.014. As compared with Fig. 7, the CNT line
shifts to higher temperatures and has a maximum as a function of φ P . The CNT and NpNT lines meet at a critical end
point (CE), which is indicated by the open circle. Such CE has
been theoretically predicted in cholesteric liquid crystals.24
Below CE temperature we find the first-order cholestericparanematic transition (CpNT) inside the binodal lines. We
find the chiral coupling between components strongly affects
the CNT.
Our theory demonstrates that the phase behaviors in binary mixtures of a polymer and a cholesteric liquid crystal
strongly depend on the external fields. When hL = 0, weonly have the coexistence region Ch+I on the temperatureconcentration plane, although, we find various phase separations in the presence of external fields. The CNT and NpNT
can be controlled by the strength of the external field and themolecular weight of a polymer.
IV. SUMMARY

We have presented a mean field theory to describe phasebehaviors in binary mixtures of a polymer and a cholestericliquid crystal in the presence of an external magnetic or electric field.35 We have calculated the orientational order parameters, the pitch of the cholesteric phase, the critical fields
for twist-untwist transitions, and the phase diagrams on thetemperature-concentration plane. It is found that in the pres-ence of the external field the CNT and NpNT take place in the
mixtures, depending on temperature and concentration. The
CNT exists at lower temperatures and at lower concentrationsof the NpNT line. On increasing the strength of the external
field, the CNT temperature shifts to lower temperatures andthe nematic phase appears between the pN and Ch phases.
In the presence of the external field, we find the various
phase separations: Ch+pN, N+pN, N1 +N2 , and Ch+N, although we only have the Ch+I coexistence in the absence ofthe external field. It is also found the triple point where threephases Ch+N+pN coexist. Due to the anisotropic coupling
between a polymer and a liquid crystal, the phase behaviors
drastically change depending on the length of a polymer. For
the strong coupling, the critical point appears at the end of
the NpNT curves. We demonstrate that a strong coupling between a polymer and a liquid crystal under the external field
takes place various phase separations.
Our theory takes into account both orientational order parameters SP and SL . For mixtures of a liquid crystal and a flexible polymer chain,16 we can take SP = 0: ν LP = 0 and cLP= 0. In this paper, we did not show the phase diagrams for the
mixtures of a cholesteric liquid crystal and a flexible polymer, however, the phase diagrams calculated with SP = 0 are
qualitatively the same as Figs. 5 and 6.45
Field-induced cholesteric-nematic phase transitions havebeen observed in a mixtures of cholesterol chloride,
nonanoate, and oleyl carbonate at a field of 3–5× 105 V/m.5 It
has been shown that CNT temperature decreases with increasing the applied field and a nematic phase appears for wide
temperature ranges. Our results of the CNT lines are consistent with these experimental results. Although the phase
diagrams calculated in this paper have not been experimentally observed yet, we believe that mixtures of a polymer anda cholesteric liquid crystal under an external field have various phase separations and provide novel materials for liquid crystalline composites and biological liquid crystals.46, 47
The problems of the boundary conditions in a cell of finite
thickness26 can also be important in these mixtures.
ACKNOWLEDGMENTS

This work was supported by Grant-in-Aid for Scientific
Research (C) (Grant No. 26400435) from the Ministry of Education, Culture, Sports, Science and Technology of Japan.
APPENDIX: MIXTURES OF A NEMATIC LIQUID
CRYSTAL AND A CHIRAL DOPANT

When ULL, 1 = 0 in Eq. (4), or cLL = 0 in Eq. (13), ourtheory can describe mixtures of a non-chiral nematic liquid
crystal (i = L) and a chiral dopant (i = P). In this appendix, we
focus on the critical value of the external field. The cholestericpitch (Q∗ ) in the absence of the external field, Eq. (14), is
given as a function of the volume fraction φ P of the chiraldopantS2n αx SP φP
L
 ,

(A1)
Q∗ =
S
1 − 1 − 2n SP φP
L

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Akihiko Matsuyama

J. Chem. Phys. 141, 184903 (2014)
10 P. Oswald, in Phase Transitions: Applications to Liquid Crystals, Organic

and we also have Eq. (28)S2n αx SP φP
L
 ,

Q∗1 =
S
1 − 1 − n SP φP

(A2)

L

where α x (≡ cLP /ν LP ) shows the chiral coupling between a liquid crystal and a dopant ( n α x = cLP /ν LL ). The critical fieldhcL is given by Eq. (46).
1, Eq. (A1) is given by
When φ P
Q∗  HT P φP ,

(A3)

SP
SL

(A4)

where
HT P ≡ 2n αx

is called the helical twisting power of a chiral dopant.1, 35, 41
The larger 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, consisting withthe experiments.41–44 The temperature and concentration dependences of HTP are given through the ratio SP /SL . Some estimations of the parameters have been studied in our previous paper.35 Using Eq. (A3), the critical field (46) is approximately given by
Hc 

π
H φ
2 TP P

K22
,
χL

(A5)

which is a linear function of φ P with SL  1. Our theory is inagreement with the experimental results.6, 9 The critical field
increases with increasing the helical twisting power and thetwist elastic constant.
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