October 20, 2017

11:18

Liquid Crystals

amLC1705˙Rev2

Liquid Crystals
Vol. 00, No. 00, Month 200x, 1–26

ArticlesTwist-bend nematic phases in binary mixtures of banana-shaped
liquid crystalline molecules
Akihiko Matsuyama∗
Department of Bioscience and Bioinformatics, Faculty of Computer Science and Systems
Engineering, Kyushu Institute of Technology, Kawazu 680-4, Iizuka, Fukuoka 820-8502,
JAPAN(May. 2017)
We present a mean field theory to describe twist-bend nematic (NT B ) phases in binary mixtures of banana-shaped, such as bent-core or dimer, liquid crystalline (LC) molecules with
K22 > K33 , where K22 is a twist and K33 is a bend elastic constant. We introduce the freeenergy for the mixtures by taking into account the director-pitch (DP) coupling γi of theconsisting LC molecules i(= A, B) and calculate the phase diagrams on the temperatureconcentration plane. Our theory demonstrates that the isotropic (I), or para-nematic (pN ),nematic (N ), and NT B phases appear, depending on the strength of the DP coupling parameter γi and predicts various phase transitions: the first-order N − pN (I) and NT B − pN (I)transitions, and the second-order NT B − N transition, etc, on the temperature-concentration
plane. Our theory has a good agreement with the experimental phase diagrams of the mixtures.
Keywords: twist-bend nematic; cholesteric; chiral nematic; nematic; banana-shaped
molecules; chirality

1.

Introduction

Recently, a new class of nematic (N ) phases has been discovered in liquid crystalline(LC) dimers and bent-core molecules (or banana-shaped molecules). The novel
twist-bend nematic (NT B ) phase is an oblique helicoidal structure with a nanoscale
pitch. It has been theoretically predicted almost 50 years ago by Meyer(1), but
only recently was observed experimentally.(2–22) It was first identified for 1, 7bis-4-(4-cyanobiphenyl) heptane, CB7CB,(3) and the NT B phase was confirmedby freeze fracture transmission electron microscopy (FFTEM)(5) , NMR,(6) and
X-ray scattering, etc.(7) After that, many studies have been conducted and theworkshop on the NT B phase has been held in 2016.(23) The research on the NT B
phase is getting active more and more.(24, 25)
The NT B phase in achiral molecules, such as banana-shaped molecules, appears
at the lower temperatures of a N phase(6, 7) and has the same director as an
oblique helicoidal structure, which is induced by electric fields applying parallel tothe pitch axis of a chiral nematic (N ∗ ) phase.(26–28) It has been suggested that the
spontaneous formation of such chiral systems from the achiral LC molecules is dueto the conformational chirality within each molecules.(29–32) The NT B phase is∗ Corresponding author. Email: matuyama@bio.kyutech.ac.jp

ISSN: 0267-8292 print/ISSN 1366-5855 online
c 200x Taylor & Francis⃝
DOI: 10.1080/0267829YYxxxxxxxx
http://www.informaworld.com

October 20, 2017

11:18
2

Liquid Crystals

amLC1705˙Rev2
Akihiko Matsuyama

Figure 1. (color online) (a) Schematic representation of the director n of the NT B phase with the coneangle ϵ between the director and the pitch axis direction p//z: p and n are unit vectors parallel to the
local director and to the helical axis, respectively, and the polarizing vector m = p × n. The angle ω isthe azimuthal angle around the z axis. The figure shows one of the two degenerate helicoidal twists (right(ω < 0) and left-handed (ω > 0) helices). We have the N ∗ phase with ϵ = π/2 and the N phase with ϵ = 0.
When 0 < ϵ < π/2, the NT B appears.(b) Banana-shaped molecule of length L and diameter D with themolecular orientation vector Ω.

mainly formed in pure LC dimers having odd spacers(3, 7) and in binary mixturesof LC dimers.(17–22) Recent experiments for the binary mixtures of LC dimers
have shown that the NT B phase is formed directory from an isotropic (I) phase
via a strong first-order phase transition,(20) in which the NT B phase depends on
not only temperature but also concentration. The motivation of this paper is totheoretically describe these binary mixtures of LC dimers.
In the NT B phase, the director exhibits periodic twist and bend deformations
forming an oblique heliconical helix with doubly degenerate domains having opposite handedness. Figure 1 shows a schematic representation of the director n of the
NT B phase. When the helix axis p is parallel to the z axis, the director is uniformly
twisted along the z axis with the pitch length p = 2π/q, maintaining a constantcone angle 0 ⩽ ϵ ⩽ π/2 with the helix axis, and is given byn(z) = (sin ϵ cos ω(z), sin ϵ sin ω(z), cos ϵ),(1)where the azimuthal angle ω is given as a function of the position z: ω = qz. The
cone angle ϵ between the director and pitch axes is a constant and does not dependon the position z. Equation (1) can also describe a N phase when ϵ = 0 and a N ∗phase when ϵ = π/2.
To describe the NT B phase in achiral systems, Dozov(33) has shown that thebend elastic constant K33 become a negative value in bent-core LC molecules.
Shamid et.al(34, 35) have also found that the bend-polarization coupling leads toa negative K33 . On the other hand, a quadratic elastic theory with a coupling
between a nematic director and an extra director field parallel to the pitch axisof a NT B phase leads to a non-negative elastic constant.(36–39) These theories
demonstrate that the twist elastic constant K22 is always K22 > K33 to deriveOctober 20, 2017

11:18

Liquid Crystals

amLC1705˙Rev2
Twist-bend nematic in binary mixtures

3

a stable NT B phase. Molecular field approaches have also been proposed as generalized Maier-Saupe theory,(40–43) as well as Onsager-like theory.(44) Recently,
we have presented the mean field theory to describe the NT B phase for a purebanana-shaped molecule, in which we take into account the coupling between the
director and pitch axis: director-pitch (DP) coupling, and described the NT B phasedepending on temperature and the coupling strength.(39) Our theory is consistent
with the elastic theory for the NT B .(36–38) However, the phase diagrams for thebinary mixtures of LC dimers showing the NT B phase have not been theoretically
discussed, yet.
In this paper, we present a mean field theory to describe the NT B phase inbinary mixtures of banana-shaped LC molecules with K22 > K33 . Based on ourprevious theory(39) of the NT B for a pure LC molecule, we extend it to the binary
mixtures, in which we take into account the DP coupling. We derive the free energyof the binary mixtures of LC dimers and explore the NT B phase, depending on
temperature and concentration. We calculate the orientational order parameter Siof the LC molecule i(= A, B) and the order parameter y(≡ sin2 ϵ) of the NT B phaseas a function of the temperature and concentration. We also calculate the phase
diagrams on the temperature-concentration plane, depending on the strength of
the DP coupling, and find a various phase diagrams: the second-order N − NT B
phase transition and the first-order pN (I) − N and pN (I) − NT B phase transitions,etc. The theory shows good agreements with the experimental phase diagrams.
In Section 2, we introduce the free energy to describe the NT B phase in thebinary mixtures. In Section 3.1, we discuss phase transitions of the NT B phasein the framework of a Landau expansion in a series of the order parameter y of
the free energy. We show the numerical results of the phase diagrams in Section
3.3 and the comparisons of our theory with the experimental phase diagrams in
Section 3.4.

2.

Free energy of a binary mixture of LC molecules with chiral interactions

In this section, we introduce the free energy of a binary mixture of low-molecularweight LC molecules with chiral interactions.
Consider a binary mixture of LC molecules A and B. Let NA be the number ofthe LC molecule A of the length LA and diameter D and NB be the number ofthe LC molecule B of the length LB and diameter D. Let χi be the bend angle
between two bent-core mesogens of the molecule i. We assume that the bananashaped LC molecule is like a rodlike molecule with the length Li = 2D tan(χi /2)
and the diameter D. We then have the ratio ni ≡ Li /D = 2 tan(χi /2). Note that
we here consider π/2 ≪ χi ≪ π for the bend angle. The volume of the LC molecule
A and that of B are given by vA = a3 nA and vB = a3 nB , respectively, where we
define a3 ≡ (π/4)D3 ≃ D3 . Let ϕA = vA NA /V and ϕB = vB NB /V be the volume
fraction of the molecule A and B, respectively, where V is the volume of the system:
ϕA + ϕB = 1.
The free energy of the mixtures consists of the following three terms:
F = Fmix + Fani + Fco .

(2)

The first term in Equation (2) is the free energy of mixing for the binary mixture
and can be given by Flory-Huggins theory for polymer blends:(45, 46)
fmix ≡ a3 βFmix /V =ϕA
ϕB
ln ϕA +
ln ϕB + χF H ϕA ϕB ,
nA
nB

(3)

October 20, 2017

11:18

Liquid Crystals

amLC1705˙Rev2
Akihiko Matsuyama

4

where χF H is the Flory-Huggins interaction parameter between LC molecules in an
isotropic phase and β ≡ 1/kB T ; T is the absolute temperature, kB is the Boltzmann
constant. In the case of binary mixtures with a good solubility, we can take theinteraction parameter of χF H = 0.
The second term in Equation (2) shows the anisotropic free energy for liquid crystalline ordering. The configuration of a constituent molecule is characterized by its
position vector r and its orientation unit vector Ω = (sin θ cos φ, sin θ sin φ, cos θ),
defined by a polar angle θ and an azimuthal angle φ, or solid angle dΩ(= sin θdθdφ),in a fixed coordinate frame. Let fi (n(r) · Ω) be the orientational distribution function of the constituent molecule i(= A, B), where n(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 Ω. The
anisotropic part of the free energy in the second virial approximation can be givenby(46)βFani =∑∫
ρifi (n(r) · Ω) ln 4πfi (n(r) · Ω)drdΩ

i=A,B

+

∫
1 ∑
ρi ρj fi (n(r1 ) · Ω1 )fj (n(r2 ) · Ω2 )2
i,j=A,B×βUij (r1 , Ω1 ; r2 , Ω2 )dR,

(4)

where dR ≡ dr1 dr2 dΩ1 Ω2 , ρi = Ni /V is the number density of the LC moleculei. The first term in Equation (4) shows the entropy changes due to orientational
ordering and Uij is the orientation-dependent intermolecular potential between two
particles i and j (i, j = A, B). The lowest-order contributions to the interaction
potential are given in a series of the Legendre polynomials:(47–52)
Uij (r1 , Ω1 ; r2 , Ω2 ) = Uij,1 (r12 )(Ω1 × Ω2 · r̂12 )P1 (Ω1 · Ω2 )+Uij,2 (r12 )P2 (Ω1 · Ω2 ),(5)

where r12 = r2 − r1 and r̂12 (= r12 /|r12 |) is a unit vector. The potential Uij,1
shows the chiral interaction between LC molecules and the pseudoscalar term
Ω1 × Ω2 · r̂12 represents scalars coupling between orientational and spatial variables. The potential Uij,2 shows the intermolecular potential for a nematic phase,
which has been used in Maier-Saupe(53) and Onsager models.(54) As usual wehere employ a simple square well interaction potential energy with a short range
d0 (≃ D) on the order of the molecule.(51) We then define a numerical parameter cij ≡ −Uij,1 /kB T for a chiral pseudoscalar interaction parameter between LC
molecules and νij ≡ −Uij,2 /kB T (> 0) for a nematic interaction (or Maier-Saupe)
parameter. The interaction parameter νij > 0 corresponds to the orientationaldependent (Maier-Saupe) interaction parameter between the LC molecules i andj. The larger values of νij imply that the two molecules i and j prefer to be parallel
to each other and a N phase tends to be more stable in the mixture. The chiralinteraction parameter cij corresponds to the strength of the chirality between the
LC molecules i and j. The larger values of cij show stronger chirality (or twist interaction) between the LC molecules i and j and a twist distortion, or a N ∗ phase,tends to be more stable in the mixture.
Using the orientational distribution function fi (n(r) · Ω), the second rank orderOctober 20, 2017

11:18

Liquid Crystals

amLC1705˙Rev2
Twist-bend nematic in binary mixtures5(i)parameter tensor Qαβ (r) is given by (55)
∫

(3
)
1
fi (n(r) · Ω) Ωα Ωβ − δαβ dΩ
2
2
⟨3
⟩
1
= Ωα Ωβ − δαβ ,22(i)
Qαβ (r) =(6)

where Ωα is the Cartesian component of the molecular orientation vector Ω at a
position r and δαβ is the Kronecker delta function.
Using Equation (6), the average over Ω1 and Ω2 in Equation (4) yields(51)∫∫fi (n(r1 ) · Ω1 )fj (n(r2 ) · Ω2 )(Ω1 × Ω2 · r̂12 )P1 (Ω1 · Ω2 )dΩ1 dΩ24(i)= ϵαβγ r̂12,α Qβµ (r1 )Q(j)γµ (r2 ),
9

(7)

and
∫∫
fi (n(r1 ) · Ω1 )fj (n(r2 ) · Ω2 )P2 (Ω1 · Ω2 )dΩ1 dΩ2
2 (i)
(j)
= Qαβ (r1 )Qαβ (r2 ),3(8)

where ϵαβγ is Levi-Civita antisymmetric tensor of the third rank and r̂12,α is the αcomponent of the unit vector r̂12 .
We here assume that the Qαβ (r2 ) does not change appreciably over the range ofthe potential and can be expanded in the Taylor series:
Qαβ (r2 ) = Qαβ (r1 + r12 )1= Qαβ (r1 ) + rκ ∂κ Qαβ (r1 ) + rκ rµ ∂κ ∂µ Qαβ (r1 ),2(9)where rκ ≡ r12,κ is the the κ component of the position vector r12 and ∂µ = ∂/∂µ
is the first spatial derivative of the tensor order parameter (κ, µ = x, y, z).
Substituting Equation (9) into (7) and (8), the anisotropic free energy of thebinary mixtures with chiral interactions can be expressed as(39, 51)3a βFani /V =∑ ϕi ∫
i=A,B

+

ni

∑ [

i,j=A,Bfi (n(r) · Ω) ln 4πfi (n(r) · Ω)dΩ
1
2 (i)
(j)
− νij ϕi ϕj Qαβ (r)Qαβ (r)
2
3

1
1
(i)
(j)
+ νij ϕi ϕj ∂µ Qαβ (r)∂µ Qαβ (r)d20
2
9
]
1
4
(i)
− cij ϕi ϕj ϵαβγ Qµβ (r)∂α Q(j)
(r)d
0 .
µγ
2
9
It is easy to develop Equation (10) for multi-component systems.

(10)October 20, 2017

11:18

Liquid Crystals

amLC1705˙Rev2
Akihiko Matsuyama

6

The tensor order parameter of the LC molecule i is given by:(55)((i)

Qαβ (r) = Si)
3
1nα (r)nβ (r) − δαβ ,
2
2

(11)

where nα is the α(= x, y, z) component of the director n and Si is the scalarorientational order parameter of the molecule i:∫
P2 (n(r) · Ω)fi (n(r) · Ω)dΩ.

Si =

(12)

The last term in Equation (2) shows the DP coupling between the director n andthe pitch field p = (0, 0, 1) of the NT B phase. Using the tensor order parameter(Equation (11)) and the director n, it is given by(37, 39)a3 βFco /V = −∑(i)

γi ϕi pα Qαβ (r)pβ ,

i=A,B

3
= −(γA ϕA SA + γB ϕB SB )(1 − y),2(13)where pα is the α(= x, y, z) component of the pitch field p and γi is the dimensionless parameter denoted the strength of the intrinsic coupling between p and(i)Qαβ of the LC molecule i. This coupling term γi has been discussed in the elastic
theory of the NT B phase and gives rise to a local oblique twisting power.(37–39)
For larger values of γi , the director n tends to align parallel to the pitch p, namely,
the system tends to be a homogeneous N phase. This term also mathematically has
the same form with the interaction between the director and the external electricfield.(27, 28)
Substituting Equation (11) into Equations (10) and (13), we obtain the freeenergy of LC phases:(39)
FLC ≡ Fani + Fco = Fnem + Fdis ,(14)

where we have separated the anisotropic free energy into two parts. One is thenematic free energy of Maier-Saupe type(53):fnem ≡ a3 βFnem /V =∑ ϕi ∫
i=A,B

ni

fi (n(r) · Ω) ln 4πfi (n(r) · Ω)dΩ

1
2
− νA ϕ2A SA− νAB ϕA ϕB SA SB212− νB ϕ2B SB− (γA ϕA SA + γB ϕB SB ),2(15)

where we write νA ≡ νAA and νB ≡ νBB for simplicity. The other is the distortionOctober 20, 2017

11:18

Liquid Crystals

amLC1705˙Rev2
Twist-bend nematic in binary mixtures

7

free energy due to the spatial variation of the director for N ∗ and NT B phases:
1
2
fdis ≡ a3 βFdis /V = νA ϕ2A SA
gA (y, Q) + νAB ϕA ϕB SA SB gAB (y, Q)212+ νB ϕ2B SB
gB (y, Q)
2
3+ (γA ϕA SA + γB ϕB SB )y,2(16)

where we define the functions gi as a function of two order parameters: the pitch
wavenumber Q ≡ qd0 and the order parameter y ≡ sin2 ϵ of the NT B phase,)(1
1gA (y, Q) ≡ (k̃2 − k̃3 )Q2 y 2 + k̃3 Q − k̃2 QA Qy,
2
2

(17)

(1
)
1
gAB (y, Q) ≡ (k̃2 − k̃3 )Q2 y 2 + k̃3 Q − k̃2 QAB Qy,
2
2

(18)

(1
)
1
gB (y, Q) ≡ (k̃2 − k̃3 )Q2 y 2 + k̃3 Q − k̃2 QB Qy,
2
2

(19)

and

with QA ≡ cA /νA , QB ≡ cB /νB and QAB ≡ cAB /νAB (Note that the notations QA
and QAB are described as Q0 and αx in the previous paper, respectively (51)). The
values of k̃2 and k̃3 correspond to the dimensionless elastic constants of the twist(n · ∇ × n)2 and bend (n × ∇ × n)2 distortions, respectively.(39) We assume thatthese constants are normalized by the typical value (∼ 10−6 dyn) of the elastic
constant of a pure LC molecule (ϕi = 1) with Si = 1. When ϕi = 1 (i = A or B)
and y = 1, the function gi has a minimum at Q = Qi . Apparently, the value of
Qi corresponds to the dimensionless wavenumber of the N ∗ phase in the pure LC
molecule i and is given by Qi (≡ ci /νi ) = d0 q0i . When y = 0 (or a N phase), we
have fdis = 0.

2.1.

Distribution functions in an equilibrium state

The orientational distribution function fA (n(r) · Ω) of the LC molecule A andfB (n(r)·Ω) of the component B are determined by the free energy (Equation (14))
with respect to these functions: (δFLC /δfi ) = 0, under the normalization condition:∫fi (n(r) · Ω)dΩ = 1.

(20)

We then obtainfA (x) =[]1
exp ΓA P2 (x) ,
ZA

(21)

fB (x) =[]1
exp ΓB P2 (x) ,
ZB

(22)

and

October 20, 2017

11:18

Liquid Crystals

amLC1705˙Rev2
Akihiko Matsuyama

8

where we define
[
(
)
(
)
ΓA ≡ nA νA ϕA SA 1 − gA (y, Q) + νAB ϕB SB 1 − gAB (y, Q)]
3+γA (1 − y) ,
2

(23)

and
[
(
)
(
)
ΓB ≡ nB νB ϕB SB 1 − gB (y, Q) + νAB ϕA SA 1 − gAB (y, Q)]
3+γB (1 − y) .
2

(24)

The constants ZA and ZB are determined by the normalization condition (Equation (20)) as ZA = 4πI0 (SA , SB ) and ZB = 4πJ0 (SA , SB ), respectively. The functions Im and Jm are defined as:
Im (SA , SB ) ≡

∫ 1

[
]
[P2 (x)]m exp ΓA P2 (x) dx,

(25)

[
]
[P2 (x)]m exp ΓB P2 (x) dx,

(26)

0

Jm (SA , SB ) ≡∫ 1
0respectively, where m = 0, 1, 2 · · · .
Substituting Equations (21) and (22) into (12), the scalar orientational orderparameters SA and SB can be determined by the two coupled-self-consistent equations:
SA = I1 (SA , SB )/I0 (SA , SB ),(27)

SB = J1 (SA , SB )/J0 (SA , SB ).

(28)

and

Using the distribution functions (21) and (22), the free energy (Equation (14))
of the LC phases is given by()
1
2
a3 βFLC /V = νA ϕ2A SA
1 − gA (y, Q)
2()+νAB ϕA ϕB SA SB 1 − gAB (y, Q)()
12
1 − gB (y, Q)+ νB ϕ2B SB
2
ϕA
ϕB
−
ln I0 −
ln J0 .
nA
nB

(29)

When y = 0, Equation (29) results in the nematic free energy of a binary mixtureof LC molecules.(46) The total free energy (F ) is given by the sum of Equations (3)and (29) .

October 20, 2017

11:18

Liquid Crystals

amLC1705˙Rev2
Twist-bend nematic in binary mixtures

2.2.

9

Determination of the order parameter y and pitch length p

We here derive the equilibrium value y ∗ and pitch wavenumber Q∗ in a thermal
equilibrium state. The procedure is to minimize the distortion free energy fdis
(Equation (16)), with respect to y and Q: (∂fdis /∂y)Q,Si = 0 and (∂fdis /∂Q)y,Si =0. It is convenient to rewrite the distortion free energy fdis as follows:
)
1 (
2
2
+ 2ϵn ϕA ϕB SA SB + ϵnB ϕ2B SB
g(y, Q; η),fdis = νA ϕ2A SA
2(30)

where we define the distortion function:
]
[
)
(1
1
2 2
k̃3 Q − k̃2 η0 QA Q + η y,g(y, Q; η) ≡ (k̃2 − k̃3 )Q y +
2
2

(31)

with the symbols
2 + 2ϵ ϕ ϕ S S + ϵ
2 2
ϕ2A SAx A B A B
xB ϕB SB2 + 2ϵ ϕ ϕ S S + ϵ
2 2 ,ϕ2A SAn A B A B
nB ϕB SB

(32)

3(γA ϕA SA + γB ϕB SB )2 + 2ϵ ϕ ϕ S S + ϵ
2 2 ,νA (ϕ2A SA
n A B A B
nB ϕB SB )

(33)

η0 ≡

η≡

and the numerical parameters: ϵn ≡ νAB /νA , ϵnB ≡ νB /νA , ϵx ≡ cAB /cA , andϵxB ≡ cB /cA . Note that the η and η0 are given as a function of the concentrationand the orientational order parameter Si . When ϕA = 1, the value of η results inthat of the pure LC molecule A.(39) The value of η0 contains the chiral interaction parameters: ϵx and ϵxB in the numerator, and the larger values of η0 favor a
N ∗ phase. On the other hand, the value of η contains the director-pitch coupling
parameters: γA and γB , and the larger values of η favor a NT B or N phase.
Using Equation (31), we obtain the equilibrium valuey ∗ (≡ sin2 ϵ) =k̃2 η0 QA /Q∗ − k̃3,k̃2 − k̃3

(34)

for the cone angle and
√
∗

Q =±

2η
,
k̃3

(35)

for the pitch wavenumber Q∗ (∝ 1/p). The sign (+) shows the left-handed helix
with QA > 0 and the sign (−) is the right-handed helix with QA < 0. In the NT B
phases, both (+) and (−) twist appear in a sample with the same probability,
because the free energy has the same value. This result is in agreement with theexperiments(5, 11) and the elastic theory.(37) We can also derive the eﬀective
twist K22 and bend K33 elastic constant of the twist (n · ∇ × n + q0A )2 and bend(n × ∇ × n)2 distortions, respectively:(39)aK22 /kB T =

)
1(
2 2
2 2
ν A SA
ϕA + 2νAB SA SB ϕA ϕB + νB SB
ϕB k̃2 ,2(36)

October 20, 2017

11:18

Liquid Crystals

amLC1705˙Rev2
Akihiko Matsuyama

10

andaK33 /kB T =

)
1(
2 2
2 2
ν A SA
ϕA + 2νAB SA SB ϕA ϕB + νB SB
ϕB k̃3 .
2(37)

The ratio of the two elastic constants is given by a constant: K22 /K33 =k̃2 /k̃3 =const.(39)The values of y ∗ and Q∗ must be a minimum of the free energy (Equation (30)).
The minimum of the free energy function fdis , or minimum of the distortion function g, which depends on y and Q variables occurs when the Hessian is positive
definite matrix:∗∗|H(y , Q )| =∂ 2 fdis ∂ 2 fdis
∂y 2 ∂y∂Q∂ 2 fdis ∂ 2 fdis∂Q∂y ∂Q21
22+ 2ϵn ϕA ϕB SA SB + ϵnB ϕ2B SB)k̃3 (k̃2 − k̃3 )Q∗2 y ∗= νA (ϕ2A SA
2
)
(
[
] K22
− 1 y ∗ > 0.
(38)
= 3k̃3 γB SB + (γA SA − γB SB )ϕA
K33
We then find the stable NT B phase appears when k̃2 > k̃3 > 0, or K22 > K33 > 0,with γA SA > 0 and γB SB > 0 because of y ∗ > 0. In this paper, we focus on SA > 0
and SB > 0.

3.

Numerical Results and Discussion

In this section we show the numerical results for the phase transitions of the NT B
phase (Sec. 3.1) and the phase diagrams (Sec. 3.2). In the following, we chosethe (+) sign (Qi > 0, Q > 0) for chirality without loss of generality. The same
discussions are also valid for the (−) sign (Qi < 0, Q < 0).

3.1.

Phase transitions between N, N ∗ , and NT B phases

In this subsection, we examine how Equation (30) describes phase transitions between NT B , N ∗ , and N phases. In the following, we set k̃2 /k̃3 (= K22 /K33 ) = 5 andk̃3 =0.1 as a typical example. The pitch wavenumber QA (= 2πd0 /p0 ) can be estimated as QA ≃ 0.1 for the typical pitch length p0 ∼10 nm and d0 = 0.5 nm. The
distortion free energy (Equation (30)) is the quadratic function of y: g = Ay 2 + By,
and can be easily understood in the framework of a Landau expansion in a series ofy. The same arguments have been used in the elastic theory for the NT B phase.(37–39)
Figure 2 shows the equilibrium value of the order parameter y (Equation (34))
as a function of η for various values of η0 . With increasing η, the order parameter
continuously decreases from 1 at η = ηCT to 0 at η = ηT N . The phase transitions
N ∗ − NT B and NT B − N are the second-order transitions because the value of theorder parameter y continuously changes. The N ∗ − NT B transition takes place aty ∗ = 1. Using Equations (34) and (35), we obtain the critical value of the N ∗ −NT B
transition:
ηCT =

k̃3
(η0 QA )2 ,2(39)

October 20, 2017

11:18

Liquid Crystals

amLC1705˙Rev2
Twist-bend nematic in binary mixtures

11

Figure 2. Equilibrium values of the order parameter y (Equation (34)) as a function of η for various valuesof η0 . With increasing η, the order parameter continuously decreases from 1 at η = ηCT to 0 at η = ηT N .

at which we have Q∗CT = ±η0 QA . The phase transition between NT B and N phases
takes place at y ∗ = 0. Using Equations (34) and (35), we obtain the critical valueof the NT B − N transition:
ηT N =(η0 QA )2 k̃22 ( k̃2 )2
=
ηCT ,
2
k̃3
k̃3

(40)

where we have Q∗T N = (k̃2 /k̃3 )η0 QA . The value of η, Equation (33), is important
to discuss the phase transitions.
We here summarize:(1)When 0 < η < ηCT , we have the N ∗ phase with y ∗ = 1 and the wavenumberof the N ∗ phase is given by
Q N ∗ = η0 Q A .

(41)(2) When ηCT < η < ηT N , the NT B phase appears. The wavenumber of the NT B
is given by Equation (35). The order parameter y ∗ of the NT B phase is given by
Equation (34).
(3)When ηT N < η, the distortion function fdis has a minimum at y = 0 and theneither N or I phase appears. When γi = 0, we have a disordered isotropic phase at
higher temperatures. When γi ̸= 0, we have a para-nematic (pN ) phase with verysmall orientational order Si and then the pN phase is almost an I phase. In thisregard, we may write pN (I) instead of pN .
The values of η and η0 depend on the concentration ϕA and the temperature
through the order parameters SA and SB . The orientational order parameters SA
and SB can be numerically solved from Equations (27) and (28).
3.2.

Phase diagrams including I, N , N ∗ , and NT B phases of a pure LC
molecule.

In this subsection, we show the phase diagram on the temperature-concentration
plane for a pure banana-shaped LC molecule i(=A or B): ϕi = 1, with ni = 4
corresponding to the bend angle χi = 127◦ .(39) We here define the temperatureparameter τi (∝ T ):(55)τi ≡ T /TN I,i = 4.55/(ni νi ),(42)

October 20, 2017

11:18
12

Liquid Crystals

amLC1705˙Rev2
Akihiko Matsuyama

Figure 3. (a) Phase diagram on the coupling strength γi and temperature T /TCI,i plane for the pure LC
molecule i(= A, B) with ni = 4 and Qi = 0.1. The solid curves show the second-order phase transitions:
N − NT B and NT B − N ∗ phase transitions. The dotted-curves show the first-order phase transitions:pN − N , pN − NT B , and pN − N ∗ phase transitions. The closed circles show the tricritical point andthe open circle shows the critical point (CP). The inset shows the phase diagram at the weak field γi .
(b) Order parameter y(= sin2 ϵ) (right) and the pitch wavenumber Q/Qi (left) of the NT B phase plottedagainst the DP coupling γi for T /TCI,i = 0.9, 0.8, and 0.75.

where TN I,i shows the N − I phase transition temperature of the pure LC moleculei. For the numerical calculations, we use νi = 4.55/(ni τi ).
Figure 3(a) shows the phase diagram on the DP coupling parameter γi (i = A, B)
and the reduced-temperature T /TCI,i plane for the pure LC molecule of ni = 4
and Qi = 0.1.(39) The temperature is normalized by the N ∗ − I phase transition temperature TCI,i at γi = 0 and then the N ∗ − I transition takes placeat τi (≡ T /TCI,i ) = 1.(51) The solid curves show the second-order phase transitions: N − NT B and NT B − N ∗ transitions. The dotted-curves show the first-orderphase transitions: pN − N , pN − NT B , and I(pN ) − N ∗ transitions. The inner
phase diagram is an enlarged view for small values of γi . Note that the pN phasehas a very small orientational order for small γi and then it is experimentally an
isotropic phase. The orientational order parameters Si jump at the first-order phasetransition points. The open circle shows the critical point (CP), which has been
discussed in nematic liquid crystals under an external field(56, 57) and surfacealigned nematic films.(58, 59) We also find two tricritical points (TCPs), closed

October 20, 2017

11:18

Liquid Crystals

amLC1705˙Rev2
Twist-bend nematic in binary mixtures

13

circles, at which the second-order phase transition line meets the first-order one
and three phases coexist. We find that the NT B phase exists between the N andN ∗ phases and always appears at the lower temperatures of the N phase. This isconsistent with the experiment of achiral systems.(7) The upper critical field γih of
the N − NT B phase transition increases with decreasing temperature. The lower
critical field γil slightly increases with decreasing temperature and the NT B appears between the N ∗ and pN phases as a function of the temperature. The NT B
phase always appears at the higher temperatures of the N ∗ phase with small γi ,although it has not experimentally observed yet. When the coupling strength γi isvery small, we have the first-order pN − N ∗ phase transition at T /TCI,i ≃ 1. Whenγil (≃ 0.0001) < γi < γih (≃ 0.0017) between two TCPs, we have the first-orderpN − NT B phase transition. As the coupling γi increases, the temperature of thefirst-order pN − N transition slightly increases toward the CP temperature. Whenthe value of γi is larger than that of the CP, the orientational order parameter
Si continuously changes and the first-order pN − N transition disappears.(39) As
the value of ni increases, the NT B phase shifts to the higher temperatures and thevalue of the upper critical field γih increases. Figure 3(b) shows the order parametery(= sin2 ϵ), or cone angle, and the pitch wavenumber Q/Qi of the NT B phase as afunction of the director-pitch coupling γi for T /TCI,i = 0.9, 0.8, and 0.75. The valueof y decreases with increasing the coupling strength γi and the pitch wavenumber
increases. For the pure LC molecules ϕA = 1, we have η0 = 1 in Equation (32).
Using Equation (40), the pitch wavenumber at the N ∗ − NT B transition is given
by Q∗T N /QA = (k̃2 /k̃3 )=5, where we have y = 0.
In this paper, we do not take into account chiral smectic C (Sc∗ ) phases of achiral
LC molecules.(60) The N ∗ phase appeared at the weak coupling γi may be theSc∗ phases. However, it can be understood that most of the banana-shaped LC
molecule capable to form the NT B phase has a nonzero γi (> γil ).
3.3.

Phase diagrams for binary mixtures of banana-shaped LC molecules.

In this subsection, we show some phase diagrams of binary mixtures of bananashaped LC molecules. Before that, we summarize the numerical parameters.
Using Equation (42), we define the ratio T̃ between two transition temperaturesof the pure LC molecules:( )TN I,Bn B νB
nB
T̃ ≡
=
= ϵnB
.
(43)
TN I,An A νAnA
Then the parameter ϵnB (≡ νB /νA ) is given by()
nA
ϵnB = T̃
,
nB

(44)

and νB = ϵnB νA . We here assume that the nematic interaction νAB between thedissimilar molecules can be proportional to the square root of the product of νAand νB :(61)νAB,ν̃ ≡ √
νA νB(45)

where ν̃ is the constant characterizing the relative strength of the nematic interaction of the dissimilar LC molecules to that of the same LC molecules. The
dependence of phase behavior for nematic binary mixtures on the parameter ν̃ has

October 20, 2017

11:18

Liquid Crystals

amLC1705˙Rev2
Akihiko Matsuyama

14

been discussed in details.(61) The smaller ν̃ values imply that the N phase tends
to be more stable in their pure N phase than in the mixed state, and vice versa.
Using Equations (42), (43), and (45), the nematic interaction parameter between
the dissimilar molecules is given by√ϵn ≡ νAB /νA = ν̃ ϵnB = ν̃

√ ( )
nA
T̃
.
nB

(46)

Similarly, for the chiral interaction cAB between the dissimilar molecules, we putcABc̃ = √
,
cA cB

(47)

where c̃ is the constant characterizing the relative strength of the chiral interactionof the dissimilar LC molecules to that of the same LC molecules. We then obtainthe chiral interaction parameter√ϵx ≡ cAB /cA = c̃ ϵxB ,(48)

where the chiral parameter ϵxB is given by(
ϵxB ≡ cB /cA = ϵnB

QB
QA

)

(

nA
= T̃
nB

)(

)
QB
.
QA

(49)

Using ϵn and ϵx , we have
QAB (≡ cAB /νAB ) = (ϵx /ϵn )QA .

(50)

We here summarize the numerical parameters:• ni (≡ Li /D = 2 tan(χi /2)) is the axial ratio of the LC molecule i with thebend angle χi .
• Qi (≡ ci /νi ) is the pitch wavenumber of the LC molecule i and is on the orderof ∼0.1 (see Section 3.1). It also means the ratio between the strength of chirality
and that of nematic ordering.(39)• T̃ is the ratio between two transition temperatures of the pure LC molecules(see Equation (43)).
• ν̃ is the constant related to the nematic interaction (see Equation (45)).
• c̃ is the constant related to the chiral interaction (see Equation (47)).
• γi is the DP coupling of the LC molecule i.
In the following, we set nA = 4 (χA = 127◦ ), nB = 3.5 (χB = 120◦ ), and
QA = QB = 0.1. We show the eﬀects of the DP coupling γi on the phase diagrams.
In this paper, we focus on binary mixtures with a good solubility and then we takethe interaction parameter χF H = 0 in Equation (3). When χF H = 0, the isotropicinteraction between LC molecules A and B is zero and then there are no macroscopic phase separations in the mixture. However when χF H is non zero, the critical√√point of the phase separations is given by χcF H = ( nA + nB )2 /2nA nB .(45) WhenχF H > χcF H , the phase separation takes place in the mixtures. It will be importantin LC polymer and LC mixtures.(46)Figure 4 shows the phase diagrams on the temperature τA (= T /TN I,A ) andconcentration (ϕB ) plane with γA = 0.003 for various values of γB : γB = 0 (a),γB = 0.0001 (b), γB = 0.001 (c), γB = 0.003 (d), γB = 0.005 (e), and γB = 0.016

October 20, 2017

11:18

Liquid Crystals

amLC1705˙Rev2
Twist-bend nematic in binary mixtures

15

Figure 4. Phase diagrams on the temperature (T /TN I,A ) and concentration (ϕB ) plane with T̃ = 0.95,
ν̃ = 0.85, c̃ = 1.05, and γA = 0.003 for various values of γB : γB = 0 (a), γB = 0.0001 (b), γB = 0.001(c), γB = 0.003 (d), γB = 0.005 (e), and γB = 0.016 (f). The solid curves show the second-order phasetransitions: N − NT B and NT B − N ∗ transitions. The dotted-curves show the first-order phase transitions:pN − N , pN − NT B , and pN (I) − N ∗ transitions. Closed circles show the tricritical points (TCPs), at
which the second-order phase transition line meets the first-order one and three phases coexist.

(f). We here define T̃ (≡ TN ∗ I,B /TN I,A ) = 0.95, ν̃ = 0.85, and c̃ = 1.05. From
Equation (44), we then obtain ϵn = 0.88, ϵnB = 1.086, ϵx = 1.1, and ϵxB =1.086. The reduced-temperature T /TN I,A is normalized by the N − pN transition
temperature of the pure molecule A at γA = 0.003 (see Figure 3 (a)). The N −NT B
transition of the pure molecule A occurs at T /TN I,A = 0.913 for γA = 0.003. The
solid curves show the second-order phase transitions: N − NT B and NT B − N ∗transitions. The dotted-curves show the first-order phase transitions: pN − N ,pN −NT B , and I(pN )−N ∗ transitions. The closed circles show the tricritical points(TCPs), at which the second-order phase transition line meets the first-order one.
Figure 4(a) shows the phase diagram for γB = 0. In this case, the director-pitch
coupling of the molecule B is zero and then the I − N ∗ phase transition only takes
place, as shown in Figure 3(a), for the pure molecule B (ϕB = 1) at T̃ = 0.95.
The pN − N phase transition curve has a minimum as a function of ϕB because ofν̃ = 0.85(< 1): the nematic interaction between dissimilar molecules is weak relative
to that in the same species. The N − NT B transition curve has a maximum as afunction of ϕB at the dilute region because of c̃ = 1.05(> 1): the chiral interaction
between dissimilar molecules is strong relative to that in the same species. The
pN − N transition curve meets the N − NT B transition curve at the TCP. We also

October 20, 2017

11:18
16

Liquid Crystals

amLC1705˙Rev2
Akihiko Matsuyama

Figure 5. (color online) (a) Order parameters SA , SB , and y plotted against the temperature for various
values of ϕB =0, 0.2, 0.6, 0.9 with γB = 0.001 in Figure 4(c). (b) pitch length p/p0 plotted against thetemperature for various values of ϕB =0, 0.6, 0.9 with γB = 0.001 in Figure 4(c).

find that the pN − NT B and the NT B − N ∗ transition curves cross at the another
TCP, where pN , NT B , and N ∗ phases coexist. The first-order pN − NT B phasetransition appears and the stable NT B phase appears in the broad range on thephase diagram. Figure 4(b) shows the phase diagram for γB = 0.0001. For the pureLC molecule B (ϕB = 1), we have the first-order pN − NT B transition at τA = 0.95
and the second-order NT B − N ∗ phase transition at τA = 0.895. At τA ≃ 0.9, theNT B phase appears over all concentration ranges. As increasing the value of γB ,the NT B −N ∗ transition temperature decreases and the TCP of the N −NT B phasetransition shifts to higher concentrations (Figure 4(c)). When γB = 0.003, we havethe pN − N and N − NT B phase transitions at ϕB = 1. The N phase appears
between the pN and NT B phases and the TCP disappears (Figure 4(d) and (e)).
The similar phase diagrams have been experimentally observed.(17, 19–21) As
shown in Figure 3(a), as increasing γB , the pN − N transition temperature slightly
shifts to higher temperatures and the N − NT B transition temperature shifts tolower temperatures. For the large γB , (Figure 4(f)), we find the critical point (CP,open circle) at the end of the pN − N transition curve on the phase diagram. The
N phase becomes dominant on the phase diagram because the director favors toalign parallel to the pitch axis for a large γB .
Figure 5(a) shows the order parameters SA , SB , and y plotted against the temperature for various values of ϕB (=0, 0.2, 0.6, 0.9) with γB = 0.001 (Figure 4(c)).
We have the first-order pN − N transition, where the orientational order parameters jump as a function of temperature. The value of SA is larger than that of SB
because of nA > nB and T̃ < 1. The orientational order parameters are nonzero in
the high temperature region, which is the evidence of the pN phase that is not dis-

October 20, 2017

11:18

Liquid Crystals

amLC1705˙Rev2
Twist-bend nematic in binary mixtures

17

Figure 6. Phase diagrams on the temperature (T /TN I,A ) and concentration (ϕB ) plane with T̃ = 1.05,
ν̃ = 1.1, c̃ = 1.05, QA = QB = 0.1, and γA = 0.003 for various values of γB : γB = 0 (a), γB = 0.0001 (b),γB = 0.001 (c), γB = 0.003 (d), γB = 0.005 (e), and γB = 0.016 (f). The solid curves show the secondorder phase transitions: N − NT B and NT B − N ∗ transitions. The dotted-curves show the first-order phasetransitions: pN − N , pN − NT B , and pN (I) − N ∗ transitions. Closed circles show the tricritical points(TCPs), at which the second-order phase transition line meets the first-order one and three phases coexist.

ordered isotropic phase, while the values of Si are very small values. Then the pN
phase corresponds to the I phase observed experimentally.(17) When ϕB = 0.2, thesecond-order N − NT B transition occurs at τA = 0.923, where the order parametery of the NT B phase continuously increases from zero with decreasing temperature.
When ϕB = 0.6 and 0.9, we have the first-order pN − NT B phase transition, where
the order parameter Si and y jump at the transition temperature. Figure 5(b) shows
the pitch length p/p0 plotted against the temperature for various values of ϕB =0,0.6, 0.9 with γB = 0.001 in Figure 4(c). When ϕB = 0, the wavenumber at the
N − NT B phase transition temperature is given by Q∗T N /QA = (K2 /K3 ) = 5 (seebelow Equation (40)) and the pitch p/p0 continuously increases from 0.2(=K3 /K2 ).
When ϕB = 0.6 and 0.9, the pitch p/p0 discontinuously jumps at the pN − NT B
transition and increases with decreasing temperature.
Figure 6 shows the phase diagrams on the temperature τA (= T /TN I,A ) andconcentration (ϕB ) plane with γA = 0.003 for various values of γB : γB = 0 (a), γB =0.0001 (b), γB = 0.001 (c), γB = 0.003 (d), γB = 0.005 (e), and γB = 0.016 (f). We
here set T̃ (= TN ∗ I,B /TN I,A ) = 1.05, ν̃ = 1.1, and c̃ = 1.05. From Equation (44),
we obtain ϵn = 1.2, ϵnB = 1.2, ϵx = 1.15, and ϵxB = 1.2. When ν̃ > 1, the pN − N
transition curve has a maximum as a function of ϕB . The N − NT B transitionOctober 20, 2017

11:18
18

Liquid Crystals

amLC1705˙Rev2
Akihiko Matsuyama

Figure 7. (color online) Comparisons between the theory and the experimental phase diagram for thebinary mixtures of two LC molecules KA (molecule A) and CBF9CBF (molecule B): the open squares
denote the N − I transition and the closed squares denote the N − NT B transition.(17) The theoreticalsolid curve shows the second-order N − NT B transition and the dotted-curves show the first-order N − pN
transition with γA = 0.0036 and γB = 0.002.

temperature increases with increasing ϕB because of c̃ = 1.05(> 1) in Figures 6(a)(d). As increasing γB , the TCP shifts to higher concentrations and the N − NT B
transition appears for the pure LC molecule B. In Figure 6(d), the N phase appears
between the pN and NT B phases for all concentrations. Further increasing γB , theN − NT B transition temperature of the binary mixture decreases with increasing
ϕB (Figure 6(e)). In Figure 6(f), the N phase becomes dominant on the phase
diagram and the CP appears at the end of the pN − N transition.
Our theory predicts various phase diagrams depending on the strength of the DP
coupling γi of the LC molecule i. Some phase diagrams have been observed experimentally. In the next subsection, we compare our theory with the experimental
phase diagrams.

3.4.

Comparisons of theory and experimental phase diagram

In this subsection, we compare our theory with experimental phase diagrams forbinary mixtures of banana-shaped LC molecules. In the followings, we set QA =QB = 0.1.
Figure 7 shows the comparison between our theory (solid and dotted curves) andthe experimental phase diagram for the binary mixtures of two LC molecules KA(molecule A) and CBF9CBF (molecule B): the open squares denote the I − Ntransition and the closed squares show the N − NT B transition.(17) The experimental phase diagram is similar to Figure 4(d) and 6(d), where the N phaseappears between pN (I) and NT B phases for all concentrations. The theoreticalsolid curve shows the second-order N − NT B transition and the dotted-curve shows
the first-order pN − N transition. We here use nA = nB = 4 for the molecular parameters. The I − N and N − NT B transition temperatures of the puremolecule A are TN I,A = 348 K and TNT B N,A = 295 K, respectively. We then obtain TNT B N,A /TN I,A ≃ 0.848, which corresponds to the ratio of the two transition
temperatures at γA = 0.0036 in Figure 3(a). The I − N and N − NT B transition
temperatures of the pure molecule B are TN I,B = 359 K and TNT B N,B = 353 K,
respectively, and we set γB = 0.002 to be the best fitting. We also obtain the
numerical parameter T̃ = TN I,B /TN I,A = 1.031. The other numerical parameters
set ν̃ = 0.97 and c̃ = 1.0. The theory can quantitatively describe the experimental

October 20, 2017

11:18

Liquid Crystals

amLC1705˙Rev2
Twist-bend nematic in binary mixtures

19

Figure 8. (color online) Order parameters (a) and pitch length p (b) as a function of ϕB for various
temperatures in Figure 7. The pitch length is calculated by p = 2πd0 /Q with d0 = 0.5 nm.

phase diagram. The N − pN (I) transition curve exhibits a slight concave curvature because of ν̃ < 1 and the N − NT B transition curve linearly increases withincreasing ϕB because of c̃ = 1.0. The NT B phase is enhanced by adding the LC
dimer CBF9CBF and the N phase appears between the narrow temperature range
at ϕB = 1. As shown in Figure 3(a), the N phase between the pN and NT B phases
can be controlled by the strength of the director-pitch coupling γi . As decreasing γi ,the two transition temperatures TNT B N,i and TN I,i approach and the NT B phase appears at higher temperatures. In this mixture, we have γA (= 0.0036) > γB (= 0.002)
and the DP coupling of the KA (molecule A) is larger than that of the CBF9CBF(molecule B).
Using the same numerical parameters, we calculate the pitch length of the NT B
phase on Figure 7. Figure 8 shows the order parameters (a) and pitch length p (b)as a function of ϕB for various temperatures in Figure 7. The pitch length of the
NT B phase is calculated by p = 2πd0 /Q with d0 = 0.5 nm. When T /TN I,A = 0.9,
the NT B phase appears above ϕB > 0.29 and the orientational order parameters(a) are almost constant Si ≃ 0.6 and the pith length (b) increases with ϕB . When
T /TN I,A = 0.98, the pN phase with a small orientational order Si ≃ 0.04 appears
between two N phases and then the experimental I phase corresponds to the pN
phase in our model.
Figure 9 shows the comparison of our theory (solid and dotted curves) with theexperimental phase diagram for the binary mixtures of two LC molecules CB9CB(molecule A) and cpd8 (molecule B): the open squares denote the N −I transition,
the closed squares denote the N − NT B .(19) The theoretical solid curve shows theOctober 20, 2017

11:18

Liquid Crystals

amLC1705˙Rev2
Akihiko Matsuyama

20

Figure 9. (color online) Comparisons between the theory and the experimental phase diagram for thebinary mixtures of two LC molecules CB9CB (molecule A) and cpd8 (molecule B): the open squares denotethe N − pN (I) transition and the closed squares denote the N − NT B transition.(19) The theoretical solid
curve shows the second-order N − NT B transition and the dotted-curves show the first-order pN (I) − N
transition with γA = 0.0034 and γB = 0.003.

second-order N −NT B transition and the dotted-curve shows the first-order pN −N
transition. We use nA = 2.9 (χA = 110◦ ) and nB = 4.1 (χB = 128◦ ) due to theexperimental bend angles. The N − I and NT B − N transition temperatures of the
CB9CB (molecule A) are TN I,A = 394.5 K and TNT B N,A = 378.4 K, respectively.
We then obtain TNT B N,A /TN I,A ≃ 0.959 and use γA = 0.0034. The N − I and
NT B − N transition temperatures of the cpd8 (molecule B) are TN I,B = 426.3
K and TNT B N,B = 382.9 K, respectively, and we set γB = 0.003 for the coupling
parameters to be the best fitting. We also obtain the numerical parameter T̃ =TN I,B /TN I,A = 1.08. The other numerical parameters set ν̃ = 0.98 and c̃ = 1.0. The
theory can quantitatively describe the experimental phase diagram. The N −pN (I)transition curve exhibits a slight concave curvature because of ν̃ = 0.98 and the
N − NT B transition curve linearly increases with increasing ϕB because of c̃ = 1.0.
In this mixture, we have TNT B N,A ≃ TNT B N,B and the strength of the DP coupling
has almost the same value: γA (= 0.0036) ≃ γB (= 0.003)
Figure 10 shows the order parameters (a) and the pitch length p (b) as a functionof ϕB at T /TN I,A = 0.9, 0.92, 0.94 in Figure 8. The pitch length is calculated by
p = 2πd0 /Q with d0 = 0.5 nm. The order parameters (Figure 10(a)) are almost
constants as a function of ϕB . As decreasing temperature at a given ϕB , the orderparameters and the pitch length increase. In the NT B phase, the orientational orderparameter SB is slightly larger than SA , because of nB > nA and T̃ > 1. The orderparameter y has small values. For example, when y = 0.02, we have the cone angle
ϵ = 10◦ . The pitch length (Figure 10(b)) has a minimum value as a function of ϕB .
Using Equation (35), the pitch length of the NT B phase is approximately given by
p
≃
p◦A

√

1 − 2(1 − ϵn )ϕB + (1 − 2ϵn + ϵnB )ϕ2B(),
1 + γγBA − 1 ϕB

(51)

where we have assumed SA ≃ SB and p◦A ≡ d0 (k̃3 νA SA /6γA )1/2 is the pitch lengthof the pure LC molecule A. From Equation (46), we have ϵn = 0.856 and ϵnB =0.764. Then the pitch length has a concave curvature as a function of ϕB . For
larger values of ϵn (> 1), for example, in Figure 6(d), the pitch length has a convex
curvature as a function of ϕB . It has been observed that the pitch of the NT B phaseis about p = 8 nm in the bent molecular dimers CB7CB.(7) The orientational orderOctober 20, 2017

11:18

Liquid Crystals

amLC1705˙Rev2
Twist-bend nematic in binary mixtures

21

Figure 10. (color online) Order parameters (a) and pitch length p (b) of the NT B phase as a function ofϕB at T /TN I,A = 0.9, 0.92, 0.94 in Figure 9. The pitch length is calculated by p = 2πd0 /Q with d0 = 0.5
nm.

parameter increases from 0.41 to 0.73 with decreasing temperature in a mixture of
CB7CB and LC molecule KA.(17) Our numerical calculations are the same orderwith the experimental data.
Figure 11 shows the comparison of our theory (solid and dotted curves) withthe experimental phase diagram for the binary mixture of two LC dimers 2O.5.O2(molecule A) and 2O.3.O2 (molecule B): the open squares denote the N − I transition, the closed squares denote the N − NT B , and the open circles show the
NT B − I transition.(20) The LC dimers 2O.5.2O and 2O.3.2O have odd spacers 5
and 3, respectively. We here set nA = 4 (χA = 127◦ ) and nB = 3.7 (χB = 123◦ ).
The experiment shows that the NT B phase is formed directly from the I phase
via a strong first-order phase transition,(20) which is demonstrated in Figure 5(a).
The pure LC molecule B has the NT B − I transitions and then the experimental
NT B − I transitions correspond to the phase diagrams with the weak DP coupling γB as shown in Figure 4(b)-(c). We here use γB = 0.001 as shown in Figure 4(c) for the numerical calculation. The N − I transition and the NT B − N
transition temperature of the 2O.5.2O molecule (A) are TN I,A = 385.3 K and
TNT B N,A = 366.9 K, respectively. We then obtain TNT B N,A /TN I,A ≃ 0.952 andcan estimate γA = 0.00255 from Figure 3(a) for the best fitting. The NT B − I
transition temperature of the 2O.3.2O molecule (B) is TNT B I,B = 360.4 K and wethen obtain the numerical parameter T̃ = TNT B I,B /TN I,A = 0.9353. We also setν̃ = 0.95 and then we have ϵn = 0.96, ϵx = 0.99, and ϵnB = 1.011. The chiralparameter is changed: c̃ = 0.95, 1.0 and 1.1. The pN (I) − N transition curve has

October 20, 2017

11:18
22

Liquid Crystals

amLC1705˙Rev2
Akihiko Matsuyama

Figure 11. (color online) Comparisons between the experimental phase diagram and theory. The symbols
show the experimental phase diagram for the binary mixtures of two LC dimers 2O.5.O2 (molecule A)
and 2O.3.O2 (molecule B): the open squares denote the N − I transition, the closed squares denote the
N − NT B , and the open circles show the NT B − I transition.(20) The theoretical solid curves show thesecond-order N − NT B transition and the dotted-curve shows the first-order phase transitions: pN (I) − N
and pN (I) − NT B transitions. The chiral parameter is changed: c̃ = 0.95, 1.0 and 1.1, with γA = 0.00255
and γB = 0.001.

a concave curvature as a function of ϕB because of ν̃ < 1. The N − NT B transition for c̃ > 1 (c̃ < 1) has a convex (concave) curvature as a function of ϕB . The
theory indicates that the TCP (closed circle) shifts to lower concentrations of ϕBwith increasing the chiral parameter c̃. The theory can qualitatively describe theexperimental phase diagram. The pN (I) − NT B phase transition is the first-orderphase transition, where the orientational order parameter Si jumps and strongly
depends on the nematic interaction parameter ν̃. Then the pN (I) − NT B phasetransition curve does not depend on the chiral interaction parameter c̃. The chiral interaction parameter c̃ aﬀects the second-order N − NT B phase transition. In
this mixture, we have γA (= 0.00255) > γB (= 0.001) and the DP coupling of the2O.5.2O (molecule A) is larger than that of the 2O.3.2O (molecule B). As shownl < γ < γ h and thein Figure 3(a), the pN (I) − NT B transition exists between γBBBh
N − NT B transition appears with γA < γA . The experimental phase diagram issimilar with Figure 4(a)(b)(c), in which γA > γB .
Figure 12 shows the order parameters (a) and pitch length p (b) as a functionof ϕB at T /TN I,A = 0.9, 0.92, 0.94 with c̃ = 1 in Figure 10. The pitch length iscalculated by p = 2πd0 /Q with d0 = 0.5 nm. As decreasing temperature at a given
ϕB , the order parameters and the pitch length increase. When T /TN I,A = 0.9 and
0.92, the mixture is in the NT B phase, where the value of the order parametery and the pitch length (p) increase with increasing ϕB . When T /TN I,A = 0.94,
we have the second-order NT B − N phase transition at ϕB = 0.52, at which thevalue of y continuously changes with decreasing ϕB and the first-order N − pN (I)transition takes place at ϕB = 0.56, at which the orientational order parameters
jump (Figure 12(a)). Due to the experimental observations of the quadrupolar
splittings for the 2O.7.2O molecule, the order parameter grows from about 0.27 at
TN I to about 0.51 before the transition to the NT B and has grown to 0.56 at the
NT B phase with decreasing temperature.(3, 24)
We demonstrate that the DP coupling γi of the LC molecule becomes an important parameter to understand the phase behaviors including the NT B phase. Our
theory predicts that there are various phase diagrams, depending on the value ofγi . It is worth to point out that the Figures 4(a)(b) and 5(a)(b), including the N ∗phase, with the weak coupling γi have not been observed yet. These phase diagrams

October 20, 2017

11:18

Liquid Crystals

amLC1705˙Rev2
Twist-bend nematic in binary mixtures

23

Figure 12. (color online) Order parameters (a) and pitch length p (b) as a function of ϕB at T /TN I,A =0.9, 0.92, 0.94 with c̃ = 1 in Figure 10. As decreasing temperature at a given ϕB , the order parameters andthe pitch length increase. The pitch length is calculated by p = 2πd0 /Q with d0 = 0.5 nm.

may be observed in the mixtures of a LC dimer such as CB7CB and a LC moleculewith a N ∗ − I transition. At the strong coupling γi , the CP appears as shownin Figures 4(f) and 5(f), which also have not been observed yet. The parameterγi corresponds to the local oblique twisting power and changes from NT B to Nphase, in which the director couples with the helical axis and aﬀects the range of
the NT B phase. The experimental verification of the coupling parameter γi , or the
local oblique twisting power, is open questions.
Our theory shows good agreements with the experimental phase diagrams. Mostof the experimental phase diagrams observed so far has been fitted with γi =0.001 ∼ 0.003. These values correspond to the external electric field, applying
parallel to the helix axis direction, approximately on the order of ∼ 1 V/µm.(39)That means the strength of the director-pitch coupling corresponds to the local
electric field on the order of ∼1 mV per a pitch length (order of nm) of the NT B
phase and it may give rise a local polarization. This becomes the local twisting
power induced by the stacking parallel to the pitch axis of the banana-shaped LC
molecules with a molecular dipole moment.(22) .
The flexoelectric eﬀects may be important to the free energy of a NT B phase.
In a usual N phase, the director n is equivalent to −n and hence the electric
polarization is zero. However a splay or a bending distortion can create an electric
polarization pf which is proportional to the first-order space derivatives of thedirector n.(55) Due to the recent theoretical study based on the Frank elastic
theory,(38) the flexoelectric polarization (pf ) in the NT B phases is perpendicularOctober 20, 2017

11:18

Liquid Crystals

amLC1705˙Rev2
REFERENCES

24

to the nematic director (Equation (1)) and then the director-flexoelectric couplingis zero: n·pf = 0. Then the contribution to the free energy due to the flexoelectricity
may be given by the coupling −E · pf , where E is an applied electric field. The
coupling parameter γi between the director and helical axis is aﬀected by theexternal magnetic or electric field applied parallel to the helical axis.(28, 38) Our
theory can be extended to the NT B phase including the flexoelectricity under theexternal fields.

4.

Summary

We have demonstrated that our theory is capable of predicting a variety of phase
diagrams for binary mixtures of banana-shaped LC molecules, in which the DP
coupling (−γi (n · p)2 ) can be an important role for the spontaneous formation of
the NT B phase. The stable NT B phase appears in the mixtures of K22 > K33 .
The strength γi of the DP coupling for the LC molecule can aﬀect the NT B andphase diagrams. Physically, this parameter corresponds to a local oblique twisting
power of the NT B phase. For the weak coupling parameters, we have the firstorder pN (I) − NT B transition with the TCP on the temperature-concentration
plane (Figure 4(a)-(c) and 5(a)-(c)). Note that the pN phase has a very small
orientational order and then it is realistically the I phase. Further increasing γi ,the NT B phase appears at lower temperatures of the N phase on the phase diagrams(Figure 4(d)(e) and 5(d)(e)). At the strong coupling γi , the NT B phase disappears,
or shifts to lower temperatures, and the first-order pN (I) − N transition appears
with the CP (Figure 4(f) and 5(f)). The parameters ν̃ and c̃ relate the shape of
the pN (I) − N and N − NT B transition curves on the phase diagram, respectively.
When c̃ > 1 (c̃ < 1), the N −NT B transition curve has a convex (concave) curvature
as a function of the concentration.
Our theory has a good agreement with the experimental phase diagrams of thebinary mixtures. The order parameters and pitch length of the NT B phase are
predicted as a function of the concentration for the experimentally observed phase
diagrams. We have shown that most of the experimental phase diagrams can theoretically be reproduced with the values of γi = 0.001 ∼ 0.004. These values
correspond to the realistic values of the local oblique twisting power for the NT B
phase. Our theory can also be applicable to study phase separations in such binary
mixtures.

References(1) Meyer RB. Structural problems in liquid crystal physics, Les Houches Summer School in Theoretical
Physics. In: Balian R, Weil G, editors. 1973 Molecular Fluids. New York (NY): Gordon and Breach:
1976. p. 273-373.
(2) Gortz V, Southern C, Roberts NW, Gleeson HF, Goodby JW. Unusual properties of a bent-coreliquid-crystalline fluid. Soft Matter. 2009; 5: 463-471.
(3) Cestari M, Diez-Berart S, Dunmur DA, Ferrarini A, de la Fuente MR, Jackso, DJB, Lopez DO,
Luckhurst GR, Perez-Jubindo MA, Ricardson RM, Salud J, Timimi BA, Zimmermann H. Phase
behavior and properties of the liquid-crystal dimer 1”, 7”-bis(4-cyanobiphenyl-4’-yl) heptane: A twistbend nematic liquid crystal. Phys Rev E 2011: 84: 031704.
(4) Henderson PA, Imrie CT. Methylene-linked liquid crystal dimers and the twist-bend nematic phase.
Liq Cryst. 2011; 38: 1407.
(5) Chen D, Porada JH, Hooper JB, Klittnick A, Shen Y, Tuchband MR, Korblova E, Bedrov D, Walba
DM, Glaser MA, Maclennan JE, Clark NA. Chiral heliconical ground state of nanoscale pitch in anematic liquid crystal of achiral molecular dimers. PNAS. 2013; 110: 15931-15936.
(6) Beguin L, Emsley JW, Lelli M, Lesage A, Luckhurst GR, Timimi BA, Zimmermann H. The Chiralityof a Twist-Bend Nematic Phase Identified by NMR Spectroscopy. J Phys Chem B. 2012; 116: 79407951.
(7) Borshch V, Kim YK, Xiang J, Gao M, Jákli A, Panov VP, Vij JK, Imrie CT, Tamba MG, Mehl GH,

October 20, 2017

11:18

Liquid Crystals

amLC1705˙Rev2
REFERENCES

25

Lavrentovich OD. Nematic twist-bend phase with nanoscale modulation of molecular orientation.
Nat Commun. 2013; 4: 2635.
(8) Sepelj M, Lesac A, Baumeister U, Diele S, Nguyen HL, Bruce DW. Intercalated liquid-crystalline
phases formed by symmetric dimers with an α, ω-diiminoalkylene spacer. J Mater Chem 2007: 17:1154-1165.
(9) Greco C, Luckhurst GR, Ferrarini A. Enantiotopic discrimination and director organization in thetwist-bend nematic phase. Phys Chem Chem Phys. 2013; 15: 14961-14965.
(10) Meyer C, Luckhurst GR, Dozov I. Flexoelectrically Driven Electroclinic Eﬀect in the Twist-Bend
Nematic Phase of Achiral Molecules with Bent Shapes. Phys Rev Lett. 2013; 111: 067801.
(11) de Almeida RRR, Zhang C, Parri O, Sprunt SN, Jákli A. Nanostructure and dielectric properties ofa twist-bend nematic liquid crystal mixture. Liq Cryst. 2014; 41: 1661-1667.
(12) Chen D, Nakata M, Shao R, Tuchband MR, Shuai M, Baumeister U, Weissflog W, Walba DM,
Glaser MA, Maclennan JE, Clark NA. Twist-bend heliconical chiral nematic liquid crystal phase ofan achiral rigid bent-core mesogen. Phys Rev E. 2014; 89: 022506.
(13) Challa PK, Borshch V, Parri O, Imrie CT, Sprunt SN, Gleeson JT, Lavrentovich OD, Jákli A.
Twist-bend nematic liquid crystals in high magnetic fields. Phys Rev E. 2014; 89: 060501(R).
(14) Yun CJ, Vengatesan MR, Vij JK, Song JK. Hierarchical elasticity of bimesogenic liquid crystals with
twist-bend nematic phase. Appl Phys Lett. 2015; 106: 173102.
(15) Hoﬀmann A, Vanakaras AG, Kohlmeier A, Mehl GH, Photinos DJ. On the structure of the Nx phaseof symmetric dimers: inferences from NMR. Soft Matter. 2015; 11:850-855.
(16) Paterson DA, Gao M, Kim YK, Jamali A, Finley KL, et.al. Understanding the twistbend nematic phase: the characterisation of 1-(4-cyanobiphenyl-4’-yloxy)-6-(4-cyanobiphenyl-4’yl)hexane(CB6OCB) and comparison with CB7CB. Soft Matter. 2016; 12: 6827-6840.
(17) Adlem K, Copic M, Luckhurst GR, Mertelj A, Parri O, Richardson RM, Snow BD, Timimi BA,
Tuﬃn RP, Wilkes D. Chemically induced twist-bend nematic liquid crystals, liquid crystal dimers,
and negative elastic constants. Phys Rev E. 2013; 88: 022503.
(18) Mandle RJ, Davis EJ, Voll CCA, Archbold CT, Goodby JW, Cowling SJ. The relationship between
molecular structure and the incidence of the NT B phase. Liq Cryst. 2015; 42: 688-703.
(19) Mandle RJ, Archbold CT, Sarju JP, Andrews JL, Goodby JW. The dependency of nematic andtwist-bend mesophase formation on bend angle. Sci Rep. 2016; 6: 36682.
(20) Dawood AA, Grossel MC, Luckhurst GR, Richardson RM, Timimi BA, Wells NJ, Yousif YZ. On thetwist-bend nematic phase formed directory from the isotropic phase. Liq Cryst. 2015; 43: 2-12.
(21) Archbold CT, Davis EJ, Mandle RJ, Cowling SJ, Goodby JW. Chiral dopants and twist-bend nematic
phase-induction of novel mesomorphic behaviour in an polar bimesogen. Soft Matter. 2015; 11: 75477557.
(22) Abberley JP, Jansze SM, Walker R, Patterson DA, Henderson PA, Marcelis ATM, Storey JMD, Imrie
CT. Structure-property relationships in twist-bend nematogens: the influence of terminal groups. Liq
Cryst. 2017; 44: 68-83.
(23) D Alessandro G, Luckhurst GR, Sluckin TJ. Twist-bend nematics and beyond, Liq Cryst. 2017; 44:
1-3.
(24) Dawood AA, Grossel MC, Luckhurst GR, Richardson RM, Timimi BA, Wells NJ, Yousif YZ, Twistbend nematics, liquid crystal dimers, structure-property relations, Liq Cryst. 2017; 44: 106-126.
(25) Panov VP, Vij JK, Mehl GH, Twist-bend nematic in cyanobiphenyls and difluoroterphenyls, Liq
Cryst. 2017; 44: 147-159.
(26) Xiang J, Shiyanovskii SV, Imrie C, Lavrentovich OD. Electrooptic Response of Chiral Nematic Liquid
Crystals with Oblique Helicoidal Director. Phys Rev Lett. 2014; 112: 217801.
(27) Matsuyama A, Field-induced oblique helicoidal cholesteric phases in mixtures of chiral and achiralliquid crystalline molecules. Liq Cryst. 2017; DOI: 10.1080/02678292.2017.1323239.
(28) Matsuyama A, Theory of a helicoidal cholesteric phase induced by an external field. Liq Cryst. 2016;
43: 783-795.
(29) Thisayukta J, Nakayama Y, Watanabe J. Eﬀect of chemical structure on the liquid crystallinity ofbanana-shaped molecule. Liq Cryst. 2000; 27: 1129-1135(30) Thisayukta J, Niwano H, Takezoe H, Watanabe J. Enhancement of Twisting Power in the Chiral
Nematic Phase by Introducing Achiral Banana-Shaped Molecules. J Am Chem Soc. 2002; 124: 33543358.
(31) Lubensky TC, Radzihovsky L. Theory of bent-core liquid-crystal phases and phase transitions. Phys
Rev E. 2002; 66: 031704.
(32) Lansac Y, Maiti PK, Clark NA, Glaser M. A. Phase behavior of bent-core molecules. Phys Rev E.
2003; 67: 011703.
(33) Dozov I. On the spontaneous symmetry breaking in the mesophases of achiral banana-shaped
molecules. Europhys Lett. 2001; 56: 247-253.
(34) Shamid SM, Dhakal S, Selinger JV. Statistical mechanics of bend flexoelectricity and the twist-bendphase in bent-core liquid crystals. Phys Rev E. 2013; 87: 052503.
(35) Shamid SM, Allender DW, Selinger JV. Predicting a Polar Analog of Chiral Blue Phases in Liquid
Crystals. Phys Rev Lett. 2014; 113: 237801.
(36) Virga EG. Double-well elastic theory for twist-bend nematic phases. Phys Rev E. 2014; 89: 052502.
(37) Barbero G, Evangelista LR, Rosseto MP, Zola RS, Lelidis I. Elastic continuum theory: Towards
understanding of the twist-bend nematic phases. Phys Rev E. 2015; 92: 030501(R).
(38) Zola RS, Barbero G, Lelidis I, Rosseto MP, Evangelista LR. A continuum description for cholestericand nematic twist-bend phases based on symmetry considerations. Liq Cryst. 2017; 44: 24-30.
(39) Matsuyama A, Director-pitch coupling-induced twist-bend nematic phase. J Phys Soc Jpn. 2016; 85:
114606.
(40) Vanakaras AG, Photinos DJ. A molecular theory of nematic-nematic phase transitions in mesogenic
dimers Soft Matter. 2016; 12: 2208-2220.
(41) Greco C, Luckhurst GR, Ferrarini A. Molecular geometry, twist-bend nematic phase and unconven-October 20, 2017

11:18
26

Liquid Crystals

amLC1705˙Rev2
REFERENCES

tional elasticity: a generalised Maier-Saupe theory. Soft Matter. 2014; 10: 9318-9323.
(42) Ferrarini A. The twist-bend nematic phase: molecular insights from a generalised Maier-Saupe theory.
Liq Cryst. 2017; 44: 45-57.
(43) Tomczyk W, Pajak G, Longa L. Twist-bend nematic phases of bent-shaped biaxial molecules. Soft
Matter. 2016; 12: 7445-7452.
(44) Greco C, Ferrarini A. Entropy-driven chiral order in a system of achiral bent particles. Phys Rev
Lett. 2015; 115: 147801.
(45) Flory PJ. Principles of polymer chemistry. Ithaca: Cornell University; 1953.
(46) Matsuyama A. Thermodynamics of flexible and rigid rod polymer blends. In: Isayev AI, editor.
Encyclopedia of polymer blends. Vol. 1. Weinheim (Germany): WILEY-VCH; 2010. Chapter 2.
(47) Luckhurst GR, Gray GW. Ed. The Molecular Physics of Liquid Crystals. Academic Press, New York,
1979.
(48) Lin-Liu YR, Shih YM, Woo CW. Molecular theory of cholesteric liquid crystals and cholestericmixtures. Phys. Rev. A 15, 2550 (1977).
(49) Lin-Liu YR, Shih YM, Woo CW, Tan HT. Molecular model for cholesteric liquid crystals. Phys. Rev.
A 14, 445 (1976).
(50) Goossens WJA. A Molecular Theory of the Cholesteric Phase and of the Twisting Power of Optically
Active Molecules in a Nematic Liquid Crystal. Mol Cryst Liq Cryst. 1971; 12: 237-244.
(51) Matsuyama A. Theory of polymer-dispersed cholesteric liquid crystals. J Chem Phys. 2013: 139;
174906.
(52) Matsuyama A. Biaxiality of cholesteric phases in rod-like polymer solutions. Liq Cryst. 2015; 42:423-429.
(53) Maier W, Saupe A. Eine einfache molekular-statische theorie der nematischen kristallinflussigen
phase. Teil I. Z Naturforsch. 1959; 14a: 882-889.
(54) Onsager L. The eﬀects of shape on the interaction of colloidal particles. Ann NY Acad Sci. 1949; 51:
627-659.
(55) De Gennes PG, Prost J. The physics of liquid crystals. New York (NY): Oxford University Press;
1993(56) Wojtowicz PJ, Sheng P. Critical point in the magnetic field-temperature phase diagram of nematic
liquid crystals. Phys Lett A. 1974; 48: 235-236.
(57) Lelidis I, Durand G. Electric-field-induced isotropic-nematic phase transition. Phys Rev E. 1993; 48:3822-3824.
(58) Sheng P. Boundary-layer phase transition in nematic liquid crystals. Phys Rev A. 1982;26: 1610-1617.
(59) Sheng P. Phase Transition in Surface-Aligned Nematic Films. Phys Rev Lett. 1976; 37: 1059-1062.
(60) Niori T, Sekine T, Watanabe J, Furukawa T, Takezoe H. Distinct ferroelectric smectic liquid crystals
consisting of banana shaped achiral molecules. J Mater Chem. 1996; 6: 1231-1233.
(61) Chiu HW, Kyu T. Equilibrium phase behaviour of nematic mixtures. J Chem Phys. 1995; 103:
7471-7481.