Phase diagrams of mixtures of apolymer and a cholesteric liquidacrystal under an external fieldAkihiko Matsuyama（松山明彦）
Department of Bioscience and Bioinformatics, Kyushu Institute of Technology, Fukuoka, JapanA. Matsuyama, J. Chem. Phys. 141, 184903 (2014).

(i)where Qαβ (r) is the second rank orientational order parameter tensor of the molecule i.
Numerical parameters: nematic coupling ϵn = νLP /νLL and chiral coupling ϵx = cLP /cLL

Abstract
We present a mean field theory to describe phase behaviors in mixtures of a polymer anda cholesteric liquid crystal in the presence of an external magnetic or electric field.We examine twist-untwist phase transitions and phase separations in the mixtures. It is found thata cholesteric-nematic phase transition (CNT) can be induced by not only the external field
but also concentration and temperature. Depending on the strength of the external field, we
predict cholesteric-paranematic (Ch+N), nematic-paranematic (N+pN), cholesteric-nematic(Ch+N) phase separations, etc, on the temperature-concentration plane.
1. Introduction
HOLESTERIC -nematic phase transition was first studied by de Gennes[1] and Meyer.[2]C
Applying an external field perpendicular to a helical axis of a cholesteric phase, untwisting of the helix takes place with positive dielectric or diamagnetic anisotropy. We here assume

that the thickness of a sample is sufficiently large and the boundary conditions can be neglected. On increasing the external field H, the pitch p of the cholesteric phase increases
and diverge at a critical value Hc of the external field. When H > Hc, we have a nematicphase, oriented uniformly along the external field (Fig. 1). This is known as a field-induced
twist-untwist transition (or CNT), which has been used in liquid crystal displays. It is wellknown that the critical field Hc is a linear function of the concentration of dopants in the diluteregime and the pitch of a helix is inverse proportional to the concentration. There are manystudies of the CNT based on the Frank’s elastic theory,[1] however, it can not describe theconcentration dependences. To describe the concentration dependences, we need to consider binary mixtures of a liquid crystal and a polymer. Recently, developing the moleculartheory of the cholesteric phase presented by Lin-Liu et.al.,[3] we have presented the molecular theory to describe phase behaviors of binary mixtures of a polymer and a cholestericliquid crystal, by taking into account of the chiral coupling between the components.[4]zHp*p

between unlike molecules.
3. Cholesteric pitch under an external field
In order to calculate the distortion free energy due to the spatial variation of the director in acholesteric phase, we assume that the director is uniformly twisted along z axis with a pitch
p = 2π/q:n(r) = (cos θ, sin θ, 0),(6)The scalar order parameters do not depend on position r and θ is a function of z in our mean
field approximations. We here set the magnetic field H is taken along y axis:

When ∆χi > 0, the molecules are oriented along the external field.
Minimizing the free energy we obtain the pitch p as a function of concentration ϕi and orientational order parameters Si:()
pQ0
4ϕLSL + 2ϵnϕP SP== 2 K(c)E(c)
,(8)
p0
Q
ϕLSL + 2ϵxϕP SPπwhere K(c) and E(c) is the complete elliptic integral of the first and second kind, respectively,
and p0 = 2πd0/Q0 is the pitch of a pure liquid crystal in the absence of the external field.
When ϕP = 0, Eq. (8) results in the pitch of the pure liquid crystal derived by de Gennes.[1]4. Phase Diagrams
The phase diagrams on the temperature-concentration plane in the absence of an external field have been discussed in the previous paper[4] where we find the phase separation(Ch+I) between a cholesteric and an isotropic phase (see Fig.2(a)). Under an external fieldhL, the CNT takes place, depending on temperature and concentration, and then the phase
diagrams are drastically changed (see Fig.2(b) and Fig.3): Cholesteric (Ch), Isotropic (I),Nematic (N), paranematic (pN, or weak nematic) phase.

y

1.05

1.15

1.05

nP=20x

H=0

Hc<H

0<H<Hc0.95 Ch

1.10 pN

nP=201.00
T/ TCI

nhL=0.014

hL=0.014

pN

I

1.00
T/TCI

θ(z)

(7)

H = (0, H, 0)

0.95

NCh+I

nP=50

1.05

ε x=0.6

N + pNT/ TCIapN+N1.00
N

0.95

0.90

0.90

Ch + pN

N1+N20.90

Ch+NChFigure 1: Twist-untwist transition induced by an external field H. Cholesteric pitch is along zaxis and the external field is taken along y axis.
In this paper we develop our previous theory[4] to describe phase behaviors in mixtures of acholesteric liquid crystal and a polymer chain in the presence of an external magnetic or electric field. We predict the field-induced CNT and novel phase separations on the temperatureconcentration plane under an external field.
2. Free Energy

0.85
0.0

0.10.2

φP

0.3

0.4

0.5

Ch

0.85
0.0

0.1

0.2

0.3

0.4

φP

0.5

The first term in Eq. (1) is the free energy of isotropic mixing of a polymer (P) and a liquid
crystal molecule (L) and is given by Flory-Huggins theory for polymer solutions:
ϕLϕP
βFmix/V =ln ϕL +ln ϕP + χϕLϕP ,(2)nLnP
where ϕi is the volume fraction of the molecule i(= L, P ), χ is the Flory-Huggins interaction
parameter between a liquid crystal and a polymer in an isotropic phase, and β = 1/kB T .
The anisotropic part of the free energy Fani in the second virial approximation can be givenby∫∑Fani = (1/β)ρi fi(n(r) · Ω) ln 4πfi(n(r) · Ω)drdΩ

0.2

0.4

0.6

φP0.8

1.05
hL=0.01

T/TCI=0.84
SP1.00

SL
0.4pN

pN

Ch + pN

0.95

N + pN

0.90
N0.95

Q/Q0
0.2

0.2

φPSL
0.3

Ch + pN

0.85

ChSP0.1

hL=0.014
nP=101.00

T/ TCI

0.6

0.00.0nP=10NT/ TCI

Order Parameters0.8(1)

0.85
0.01.05
hL=0.014F = Fmix + Fani + Fext.

0.6

Figure 2: Phase diagram in the absence of the external field hL = 0 (a) and hL = 0.014 (b).
The solid curves are the binodal lines, the black (blue) dotted line shows the CIT (NpNT), andthe dashed line is the CNT, which corresponds to the critical field. The NpNT temperatureof the pure liquid crystal (ϕP = 0) shifts to higher temperatures because of the external field.
The CNT and NpNT lines meet at a critical end point (CEP), open circle. Below the CEP, thefirst order cholesteric-paranematic phase transition (ChpNT) appears. (c): Phase diagramfor ϵx = 0.2 and nP = 50 with hL = 0.014. The closed circles show a critical point (CP).
1.0The free energy consists of the following three terms:pN

Ch

0.4

0.5

0.90
0.00

0.10

φP

0.20

0.30

0.80
0.0

0.1

0.2

φP

0.3

0.4

0.5

Figure 3: (a) Orientational order parameters SL, SP , and the pitch wavenumber Q/Q0(∝ p−1)
plotted against ϕP for nP = 10. The second-order CNT takes place at ϕP ≃ 0.1 and the firstorder nematic-paranematic transition (NpNT) occurs at ϕP ≃ 0.32. (b) (c): Phase diagram for
ϵx = 0.2 and nP = 50 with hL = 0.01 (b) and hL = 0.014(c).

i=L,P∫∑
1
ρiρj fi(r1, Ω1)fj (r2, Ω2)Uij (r1, Ω1; r2, Ω2)dR,+2(3)i,j=L,P

where dR ≡ dr1dr2dΩ1Ω2. The first term in Eq. (3) shows the entropy changes due to an orientational ordering and Uij is the orientation-dependent intermolecular potential between twoparticles i and j (i, j = L, P ). We here take UP P = 0 because we consider non-nematic polymer chains in the constituent pure polymers. The lowest-order contributions to the interaction
potential for the cholesteric phase are given by in a series of the Legendre polynomials:[3, 4]
Uij (r1, Ω1; r2, Ω2) = Uij,1(r12)(Ω1 × Ω2 · r̂12)P1(Ω1 · Ω2)+Uij,2(r12)P2(Ω1 · Ω2),(4)where we truncated by P2(x). The potential βUij,1(= cij ) shows the chiral interaction between
two particle i and j. The term Ω1 × Ω2 · r̂12 represents scalars coupling between orientational and spatial variables, where r̂12 = (r2 − r1)/|r12|. The potential βUij,2(= νij ) shows theintermolecular potential that accounts for the formation of a nematic phase in Maier-Saupemodel.
The last term in Eq. (1) is the magnetic (or electric) free energy relevant to orientational order.
When the external magnetic field H is applied to the molecule i(= L, P ) having a diamagnetic
anisotropy ∆χi, the external free energy is given by[1]∫∑(i)niρi ∆χiHαQαβ (r)Hβ dr
Fext = −(5)i=L.P

5. Summary(1) We have presented a mean field theory to describe phase behaviors in binary mixtures ofa polymer and a cholesteric liquid crystal in the presence of an external magnetic or electricfield.
(2) It is found that in the presence of the external field the CNT and NpNT take place inthe mixtures, depending on temperature and concentration. On increasing the strength ofthe external field, the CNT temperature shifts to lower temperatures and the nematic phaseappears between the pN and Ch phases.
(3) In the presence of the external field we find the various phase separations: Ch+pN, N+pN,
N1+N2, Ch+N, and Ch+N+pN, although we only have the Ch+I coexistence in the absenceof the external field. We demonstrate that a strong coupling between a polymer and a liquid
crystal under the external field takes place various phase separations.
References[1] P. G. De Gennes, Solid State Commun. 6, 163 (1968).
[2] R. B. Meyer, Appl. Phys. Lett. 12, 281 (1968).
[3] Y. R. Lin-Liu, Y. M. Shih, C. W. Woo, and H. T. Tan, Phys. Rev. A 14, 445 (1976).
[4] A. Matsuyama, J. Chem. Phys. 139, 174906 (2013).