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European Journal of Applied Sciences – Vol. 13, No. 02
Publication Date: April 25, 2025
DOI:10.14738/aivp.1302.18168.
Hosino, M. (2025). Nematic Phase Transitions in an Assembly of Molecules with a Bent Core. European Journal of Applied Sciences,
Vol - 13(02). 153-164.
Services for Science and Education – United Kingdom
Nematic Phase Transitions in an Assembly of Molecules with a
Bent Core
Masahito Hosino
Independent researcher, Norikura 1-1303-2,
Midori-ku, Nagoya 458-0004, Japan
ABSTRACT
Approximately 20 years ago, Hosino and Nakano [1] investigated the phase
transitions in an assembly of biaxial molecules that interact via dispersion forces,
discovering the existence of one uniaxial and two biaxial nematic phases and
disconematic phase in the assembly. Follow-up studies obtained valuable results on
the phase transitions in an assembly of biaxial molecules with hard cores [2],
chirality [3, 4], and flexibility [5] based on the initially proposed method [1]. In this
study, the phase transitions in the assembly of molecules with a bent core were
investigated using the same method, demonstrating the occurrence of sequence of
transitions between various nematic phases in the system.
Keyword: bent-core molecules, temperature dependence of anisotropic parameters,
sequence of transitions between nematic phases
INTRODUCTION
Approximately 20 years ago, Hosino and Nakano [1] investigated the phase transitions in an
assembly of biaxial molecules that interact via dispersion forces, thereby noting the existence
of a uniaxial and two biaxial nematic phases and a disconematic phase.
In addition, the phase diagram revealed sequence of transitions between these phases at points
where anisotropy parameters εx and εy, which represent the anisotropy in the x and y directions
of a molecule toward the z direction, respectively, are limited. As such, subsequent studies
proposed the use of the initially proposed the analytical method [1] to investigate the phase
transitions in an assembly of biaxial molecules with hard cores [2], and a system of molecules
with a chiral plane [3, 4]. These investigations revealed the similar phase transition diagrams
in a system of molecules with hard cores and molecules interacting via dispersion forces. In
addition, the mechanism for the chiral phases in a molecular system with chiral plane was
elucidated. Recently, the method obtained by Hosino and Nakano [1] was used to analyze the
phase transitions in an assembly of flexible molecules [5]. Thus, the proposed analytical method
was demonstrated to be a useful and powerful tool for investigating biaxial liquid crystals.
Several experimental and theoretical works have focused on bent liquid crystalline molecules
over the past decades [6]. It is noticed that there are two types in regard of bent liquid
crystalline molecule. One is a molecule having a bent core with a wing at each side of its core.
Another one is dimers linked by flexible chain. Both types of molecule bend depending on
temperature and/or a density of molecules. However, different model has to be proposed for
each type of molecule, respectively. The molecular model of dimers linked by flexible chain was
already proposed and the dependence of nematic transitions on molecular shape and/or
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European Journal of Applied Sciences (EJAS) Vol. 13, Issue 02, April-2025
anisotropy of interaction was elucidated [7, 8]. In those studies, the dependence of molecular
shape and/or anisotropy of interaction on temperature is induced by the internal rotation
around the C―C bond in the alkyl chain when transforming between the gauche and trans states
[9, 10]. However, another type bent liquid crystalline molecule can’t give rise to such an internal
rotation as dimers linked by flexible chain. So then, molecular model of biaxial molecule with
bent core, which has a mechanism giving the dependence of molecular shape and/or anisotropy
of interaction on temperature, is proposed in present study. Molecular model in present study
can be expanded to the one of dimers linked by flexible chain, and such an expanded model
elucidates nematic transitions more clearly than the studies carried out until now, because the
dependence of anisotropy parameters εx and εy on temperature can be obtained accurately,
and the proposed method described in previous work [5] elucidates more clearly nematic
transitions basing on phase diagram on these anisotropy parameters plane. Furthermore, in
such an expanded model, molecular shapes including chiral one are explicitly defined
depending on temperature. This fact is very important to elucidate phase transitions among
various smectic phases as SmA, SmC, and SmC* phases. The interaction potential between a pair
of molecules based on previously developed analytical methods [1―5] and one of a system of
biaxial molecules with a bent core were similar, whereby, εx and εy are functions of θB as
follows:
εx = cos(θB⁄2)⁄sin(θB⁄2) , εy = 0. (1)
As the dependence of θB on the temperature could be estimated, (as described in Section 2), the
previously proposed approach [5] elucidated the phase transition of the assembly of molecules
with a bent core.
INTRAMOLECULAR INTERACTION AND DEPENDENCE OF θB ON TEMPERATURE
As an example, the molecule of the mesogenic compound 4,6-dichloro-1,3-phenylene bis [4-(4-
n-alkyloxy-3-fluoro-phenyliminomethyl) benzoate] [Fig.1(a) and 1(b)] was considered to be a
biaxial molecule with a bent core. The θB value of this molecule was limited to the range of 2π/3
to π. The upper limit of θB was attributed to the repulsion force between the chlorine
substituent on the phenylene group and oxygen atom of carboxyl group, whereas lower limit of
θB, was ascribed the structure of the phenylene group. In addition, dipoles induced in the
benzoate groups of either side of the molecular core and both intra-molecular and inter- molecular interactions were induced between the pairs of induced dipoles. The sides of the
molecule are denoted as sides A and B (Fig. 1(b)).
With respect to the intramolecular interaction, the pseudopotential for limiting θB is introduced
as
Upseu(cosθB) =
Upseu
Cl
cos (θB)+1
−
Upseu
0
cos (θB)+1/2
(2)
Although other functions of cos(θB) can be introduced as the pseudopotential for the limiting
on θB, Eq. (2) was conveniently adopted in this work. The intramolecular interaction caused by
the induced dipoles at both sides of the core is
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155
Hosino, M. (2025). Nematic Phase Transitions in an Assembly of Molecules with a Bent Core. European Journal of Applied Sciences, Vol - 13(02).
153-164.
URL: http://dx.doi.org/10.14738/aivp.1302.18168.
Udis(cosθB
) = −
Udis
0
|r⃗⃗⃗AB⃗⃗⃗⃗⃗ |
2n
(cosθB
)
2 = −
Udis
0
(2r0
)2n
(cosθB)
2
(sin(θB⁄2))
2n = −
Udis
0
(2r0
)2n
(cosθB)
2
(1−cosθB)n
(3)
where r⃗⃗AB⃗⃗⃗ is the distance between the induced dipoles on sides A and B and r0 is the distance
between the center of the molecule and point at which the dipole was induced. Eq. (3) was
introduced as an application of Eq. (9) in Ref. [1].
The average of quantities Q depending on cosθB is
〈Q(cosθB)〉cosθB =
∫ Q(cosθB)P(cosθB)d(cosθB)
∫P(cosθB)d(cosθB)
(4)
where P(cosθB) is a weight function with respect to cosθB, calculated as
P(cosθB) = P0exp (−
Upseu(cosθB)
Udis
̅̅̅0̅̅̅ − βUdis(cosθB)) . (5)
In Eq. (5), Upseu(cosθB) is normalized with Udis
̅̅̅0̅̅ ≡ Udis
0
(2r0
)
2n ⁄ , instead of kBT, because it is
largely independent of the temperature and imposes the limitation on cosθB. Meanwhile,
Udis(cosθB) was normalized with kBT because the interaction represented with this potential
highly depends on the temperature. Although the average cosθB, denoted as cosθB
av, could be
obtained using Eq. (4), such a calculation is difficult. Therefore, a simpler method was proposed
to obtain cosθB
av. As P(cosθB) reaches its maximum value at cosθB
av
, ∂P(cosθB)⁄∂(cosθB) = 0
:thus, following equation is deduced as
−
Upseu
Cl Udis
̅̅̅0̅̅̅ ⁄
(cosθB
av+1)
2 +
Upseu
O Udis
̅̅̅0̅̅̅ ⁄
(cosθB
av+1/2)
2 =
2Udis
̅̅̅0̅̅̅
kBT
2cosθB
av+(2+n)(cosθB
av)
2
(1−cosθB
av)
n+1 . (6)
Eq. (6) can be solved graphically as shown in Fig. 2(a). First, curve lL was drawn by plotting the
values on the left-hand sides of Eq. (6) on the x-y plane for each of cosθB value. lL was drawn
within the cosθB range from -1/2 to -1. Similarly, lR was drawn by plotting the values of the
right-hand side of Eq. (6) on the same plane for each cosθB value.
The solution of Eq. (6) is given as the intersection of lL and lR. lL intersects the x-axis at cosθB
0
where the value of the pseudopotential is minimized. Moreover, the curve lL becomes infinitely
monotonic and asymptotic near cosθB = ― 1⁄2 as cosθB increases from cosθB
0
to -1/2. As
cosθB decreases from cosθB
0
to -1, lL becomes infinitely monotonic and asymptotic near
cosθB = ― 1. In contrast, lR intersects the x-axis at cosθB = 0 and-2/(n+2), which remains
below the x-axis in the cosθB region from 0 to -2/(n+2). Above the x-axis, lR exists as an
approximately straight line. When cosθB 〈- 2/(n+2), lR has a slightly upward convex.
Hereafter, n=3, in which -2/(2+n) is equal to -0.4, was assumed. When the temperature
decreased to 0, lR became asymptotic to the cosθB = -0.4. when the temperature increased
from 0 to infinity, lR became asymptotic to the x-axis.
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European Journal of Applied Sciences (EJAS) Vol. 13, Issue 02, April-2025
Therefore, an intersection always exists between the curves for all temperatures. As the
temperature decreased to 0, cosθB increased to -1/2 and εx increased to √3
3
≈ 0.5774. As the
temperature increased from 0 to infinity, cosθB decreased to cosθB
0
and εx decreased to εx
0
, as
defined in
εx
0 = √(1 + cosθB
0
) (1 − cosθB
0 ⁄ ) . (7)
Using parameter u defined as (Upseu
Cl Upseu
O ⁄ )
1
2
, cosθB
0
and εx
0
are expressed as
cosθB
0 = −
1
2
•
u+2
u+1
, (8)
εx
0 = (
1−1⁄(1+u)
3+1⁄(1+u)
)
1
2
. (9)
Overall, εx decreases from 0.5574 to εx
0
, as the temperature increases from 0 to infinity,
determining the dependence of εx on temperature. Interestingly, the dependency of εx on
temperature depends on u, which represents the intramolecular interactions as Upseu
Cl and
Upseu
O [Fig. 2(b)]. The dependence of εx on T̅ is depicted in εx
(T̅: εx
0
) curve, where T̅ is the
normalized temperature T̅ = kBT 2Udis
̅̅̅0̅̅ ⁄ . These results of the dependence of θB on
temperature derived in this section are the reasonable and consistent with the experimental
results [10]. The results obtained in this section are discussed in Section 3.
INTERMOLECULAR INTERACTION
Subsequently, the intermolecular interaction potential between the I- and J-th molecules was
investigated, as follows:
ФIJ(RIJ) = Ф(p A
(I), p A
(J)) + Ф(p A
(I), p B(J)) + Ф(p B(I), p A
(J)) + Ф(p B(I), p B(J)) (10)
where p A
(I) is the dipole induced in the group on A side of the I-th molecule, and
Ф(p A
(I), p A
(J)) is the potential of the interaction between p A
(I) and p A
(J). p A
(I) and p B(I) are
expressed in the molecular coordinate system as(|p A
(I)|cos(θB⁄2), 0, |p A
(I)|sin(θB⁄2)) and
( |p B(I)|cos(θB⁄2), 0, −|p B(I)|sin(θB⁄2)), respectively. The direction of the long axis of the
molecule and direction of vector r⃗A + r⃗⃗⃗B are defined as the z and x directions of the molecular
coordinate system, respectively. Thus, the bent molecule considered in this has a biaxial shape
with a direction normal to the molecular plane, defined as the y direction of the molecular
coordinate system. As the intermolecular potential for the biaxial molecules, particularly, the
potential for interacting via dispersion forces was introduced in a previous work [1], the terms
in Eq. (10) can be written as follows:
Ф(p A
(I), p A
(J)) = Фr(|R⃗
IJ + r A
(J) − r A
(I)|) (sin (
θB
2
))
4
f(εx
) , (11)
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157
Hosino, M. (2025). Nematic Phase Transitions in an Assembly of Molecules with a Bent Core. European Journal of Applied Sciences, Vol - 13(02).
153-164.
URL: http://dx.doi.org/10.14738/aivp.1302.18168.
Ф(p A
(I), p B(J)) = Фr(|R⃗
IJ + r B(J) − r A
(I)|) (sin (
θB
2
))
4
f(εx
) , (12)
Ф(p B(I), p A
(J)) = Фr (|R⃗
IJ + r A
(J) − r B(I)|) (sin (
θB
2
))
4
f(εx
) , (13)
Ф(p B(I), p B(J)) = Фr (|R⃗
IJ| + r B(J) − r B(I)) (sin (
θB
2
))
4
f(εx
) , (14)
where R⃗
IJ ≡ R⃗
J − R⃗
I
, that is the vector connecting the center of the I- and J-th molecules; r A
and r B are the vectors connecting the center of the molecule and positions of the induced dipole
in the group on sides A and B, respectively; and f(εx
) is defined as
f(εx
) = ∑ Q11
αβ(I)
α,β=x,y,z
Q11
αβ(J) +
(εx
)
2
2 − (εx
)
2 ∑ [Q11
αβ(I)R
αβ(J) + Q11
αβ(J)R
αβ(I) ]
α,β=x,y,z
+
(
(εx)
2
2−(εx)2
)
2
∑ R
αβ(I)R
αβ(J) α,β=x,y,z
, (15)
where Q11
αβ(I) and R
αβ(I) are defined as follows. First, the unit vector a 1
(I) was defined to
extend along the long molecular axis; the unit vector a 2
(I) was defined to extend along the
vector r⃗A + r⃗⃗⃗B ; and a 3
(I) = a 1
(I) × a 2
(I) was established to be normal to the molecular
plane. Table I presents the relationship between the directions of the orthogonal coordinate
system (ξ,η,ζ) in the molecular frame and orthogonal coordinate system (x, y, z) of the
laboratory frame with cosines between two coordinate systems shown in terms of the Eulerian
angles (θI
,φI
, ψI) (Fig. 3). Second, according to Priest and Lubensky [11], the tensors in Eq. (15)
are defined as
Qpq
ij (I) = ap
i
(I)aq
j
(I) −
δpq δij
3
(p, q = 1,2,3; i, j = x,y,z) (16)
R
αβ(I) ≡ Q22
αβ(I) − Q33
αβ(I). (17)
In Eq. (16), ap
i
(I) denotes the i-th orthogonal component of unit vector a p
(I) parallel to the p- th principal axis of the I-th molecule, andδpq and δij are the Kronecker’s delta functions.
ФIJ(RIJ) in Eq. (10) can be written as
ФIJ(RIJ) = 4 (sin (
θB
2
))
4
Фr(|R⃗
IJ|)f(εx
) . (18)
The terms r A
(J) − r A
(I) , r B(J) − r A
(I) , r A
(J) − r B(I) , and r B(J) − r B(I) in Eqs. (11―14) are
neglected because they do not affect the orientational ordering in the nematic phase. However,
they are considered to affect the translational ordering in the smectic phase, causing the
smectic A-to-smectic C phase transition. This problem is discussed in detail in Section 5. The