Emergence of Biaxiality in Nematic Liquid Crystals with Magnetic Inclusions Some Theoretical Insights Aditya Vats1Sanjay Puri2and Varsha Banerjee1

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Emergence of Biaxiality in Nematic Liquid Crystals with Magnetic Inclusions: Some

Theoretical Insights

Aditya Vats,1Sanjay Puri,2and Varsha Banerjee1

1Department of Physics, Indian Institute of Technology Delhi, New Delhi–110016, India.

2School of Physical Sciences, Jawaharlal Nehru University, New Delhi–110067, India.

The biaxial phase in nematic liquid crystals has been elusive for several decades after its prediction

in the 1970s. A recent experimental breakthrough was achieved by Liu et al. [PNAS 113, 10479

(2016)] in a liquid crystalline medium with magnetic nanoparticles (MNPs). They exploited the

diﬀerent length-scales of dipolar and magneto-nematic interactions to obtain an equilibrium state

where the magnetic moments are at an angle to the nematic director. This tilt introduces a second

distinguished direction for orientational ordering or biaxiality in the two-component system. Using

coarse-grained Ginzburg-Landau free energy models for the nematic and magnetic ﬁelds, we provide

a theoretical framework which allows for manipulation of morphologies and quantitative estimates

of biaxial order.

I. INTRODUCTION

Liquid crystal (LC) phases are mesomorphic states be-

tween ordinary liquids and crystals. The constituent

molecules translate freely as in a liquid while exhibiting

long-range orientational order. The simplest LCs are ne-

matic liquid crystals (NLCs), where constituent particles

are often rod-like or disc-shaped. The NLC molecules

typically orient along a preferred direction ncalled the

director. They exhibit uniaxial order if the molecular

alignment is only about n. Alternatively, there can be

an additional distinguished (secondary) director k(per-

pendicular to n) for orientational ordering. These are

referred to as biaxial nematic liquid crystals (BNLCs),

and were predicted by Freiser in 1970 [1]. BNLCs have

been the subject of much experimental and theoretical

research [2–8]. They are believed to oﬀer signiﬁcantly

improved response times and better viewing characteris-

tics in displays, optical switching and optical imaging as

compared to their uniaxial counterparts [7, 8].

The working principle behind LC applications is the

Fr´eedericksz transition, where the light transmissibility

changes when the NLC molecules go from an ordered

state to a disordered state [7–10]. In BNLCs, it was

predicted that this transition could occur along more

than one direction. However, the experimental detec-

tion of thermotropic BNLCs was elusive until 2004, when

three groups independently demonstrated the existence

of the biaxial phase [11–13]. It was observed that the

Fr´eedericksz transition about the secondary director is

energetically favorable, yielding light transmission that

can potentially be switched on and oﬀ more abruptly [7–

10, 14]. These experiments also revealed that the switch-

ing time is at least an order of magnitude faster in BNLCs

(∼1 ms) as compared to uniaxial NLCs (∼15 ms) [8, 14].

Despite these major advances on the experimental side,

the biaxial phase remains a challenge because the order-

ing of molecules along the secondary director is fragile

and easily destroyed by thermal ﬂuctuations [7, 8]. So

the quest for a robust biaxial phase continues.

A breakthrough in this direction is provided by the re-

cent experiments of Liu et al., where they achieved the

elusive biaxial phase by immersing magnetic nanoparti-

cles (MNPs) in an NLC medium [15]. These fascinating

ferronematics (FNs) were ﬁrst proposed theoretically in

1970 by Brochard and de Gennes with the purpose of

enhancing the magnetic response in NLCs for magneto-

optic eﬀects [16]. Unfortunately, in experimental sam-

ples, MNPs ﬂocculated within tens of minutes due to

dipole-dipole interactions [17]. It was only four decades

later, in 2013, that Mertelj et al. designed the ﬁrst

such stable suspension using barium hexaferrite magnetic

nanoplatelets in pentylcyanobiphenyl (5CB) LCs [17, 18].

They overcame the challenges of ﬂocculation by cleverly

choosing the shape and composition of the MNPs, and a

homeotropic MNP-NLC coupling.

In their experiments with FNs, Liu et al. [15] lever-

aged the diﬀerent length-scales of dipolar and magneto-

nematic interactions to obtain an equilibrium state where

the magnetic moment of the MNPs is at an angle to the

nematic director n. Such a coupling introduced an addi-

tional direction of order (k) in the perpendicular plane

at no additional cost, see the schematic in Fig. 1. Subse-

quently, the authors conﬁrmed the presence of biaxial or-

der from the absorption spectrum and magnetic hystere-

sis studies. This development opens up newer horizons

for applications of NLCs, and these require theoretical

guidance. In this paper, we provide the requisite frame-

work to study biaxial order in FNs. We will demonstrate

how the magneto-nematic coupling introduces biaxiality

in the system, even though it is absent in the pure NLCs.

We also provide quantitative evaluations of biaxiality as

a function of the coupling strength, which will be useful

for experimentalists.

This paper is organized as follows. In Sec. II, we intro-

duce the order parameters and coarse-grained free energy

for FNs. In Sec. III, we present results for the ordering

kinetics of FNs, and the development of biaxiality. In

Sec. IV, we conclude with a summary and discussion.

arXiv:2210.03467v1 [cond-mat.soft] 7 Oct 2022

II. COARSE-GRAINED FREE ENERGY FOR

FERRONEMATICS

FNs are described in terms of two order parameters:

(i) the Q-tensor, which contains information about the

orientational order of the NLCs, and (ii) the magneti-

zation vector M, which gives the average orientation of

the magnetic moments of the MNPs. The Q-tensor is

symmetric and traceless, and is given by [19]:

Qij =Sninj+T kikj−(S+˜

T)δij

3.(1)

Here, the scalar order parameter Smeasures the uniaxial

degree of order about the leading eigenvector or the di-

rector n. Further, ˜

Tis the magnitude of the biaxial order

about the secondary director k. (A system with only uni-

axial order has ˜

T= 0. For such a system, the isotropic

phase corresponds to S= 0, and the nematic phase has

S 6= 0.) Taking into account the requirements of symme-

try and tracelessness, the Q-tensor can be expressed in

terms of ﬁve independent parameters as follows:

Q=

−q1+q2q3q4

q3−q1−q2q5

q4q52q1



.(2)

To obtain the nematic directors and S,˜

T, we choose

a frame of reference in which Qis diagonal. This pro-

vides us the three eigenvalues (λ3> λ2> λ1), and the

corresponding eigenvectors n,k,l. The largest eigen-

value λ3=S, and the corresponding eigenvector is the

primary direction of order n[19, 20]. We will use a

standard measure of biaxial order about the secondary

director k:T= (λ2−λ1)/λ3[7, 8, 20], which is pro-

portional to ˜

T. Naturally, λ1=λ2if the system is uni-

axial. The degree of biaxiality can also be deﬁned as

B2={1−6Tr(Q3)2/[Tr(Q2)3]}[21, 22], where B2= 0

for the uniaxial state and B2= 1 for a state with maxi-

mum biaxiality. This deﬁnition of biaxiality also exploits

the diﬀerence between two eigenvalues to determine bi-

axial order, similar to T.

We use the Landau-de Gennes (LdG) approach to write

down the phenomenological free energy for this compos-

ite system. This is a functional of the order parameter

ﬁelds Q(r) and M(r) and has three contributions [23–28]:

G[Q,M] = ZdrA

2Tr(Q2) + C

3Tr(Q3) + B

4[Tr(Q2)]2

2|∇Q|2+α

2|M|2+β

4|M|4+κ

2|∇M|2

−γµ0

i,j=1

Qij MiMj





.(3)

The ﬁrst four terms in Eq. (3) represent the Ginzburg-

Landau (GL) free energy for the nematic component with

Landau coeﬃcients A,B,C,Lhaving their usual mean-

ing. The next three terms correspond to the GL free

energy for the magnetic component. In the GL frame-

work, the gradient terms |∇Q|2and |∇M|2are essen-

tial to capture the eﬀects of elastic interactions [29–33].

They penalise local variations in the order parameters

– this surface tension results in the motion of domain

boundaries in coarsening kinetics.

The magnitudes of the Landau coeﬃcients determine

the scales of order parameter, length and time in the sys-

tem. For example, A=A0(T−TN) and α=α0(T−TM)

depend on the quench temperature Tand the critical

temperatures TN,TM. (Here, A0,α0are material-

dependent constants.) A direct estimate of the coeﬃ-

cients can be obtained from experimentally determined

quantities like the latent heat, order parameter magni-

tudes, susceptibilities, etc. [30, 33]. However, the cur-

rent experimental data on FNs is not adequate to pro-

vide accurate estimates of these coeﬃcients. The utility

of the LdG framework lies primarily in predicting uni-

versal behaviors, e.g., power laws and their exponents,

scaling variables, etc.

The eﬀect of dopant particles in LCs has been mod-

eled in several previous studies [34–37]. These models

describe the coupling of the dipole moment of ferroelec-

tric particles with the NLCs at a molecular level. The

induced ﬁeld due to the impurity atoms acts like an

aligning ﬁeld, and enhances orientational order in the

NLCs. On a similar footing, the last term in Eq. (3)

is the phenomenological magneto-nematic coupling de-

ﬁned as a dyadic product of Qand Mand the parame-

ter γis the strength of the coupling. It is related to the

shape and size of the MNPs and their interaction with the

NLCs. This cubic magneto-nematic coupling term [24]

enforces the speciﬁc orientations of the magnetic and ne-

matic components essential for the emergence of biaxial

order in the system [15, 38]. A more accurate description

of the free energy can be obtained by incorporating dipo-

lar and quadrupolar interactions. This may be required

for studies of phase transitions and critical phenomena.

As discussed in Ref. [26], these terms may be ignored for

dilute ferronematic suspensions.

In their experiments, Liu et al. demonstrated that bi-

axial order emerges only when nand Mare tilted at

an angle. By manipulating the surface functionaliza-

tion, they could achieve a tilt angle up to 90◦. (Their

optical absorbance measurements to detect the biaxial

phase were carried out for a limited range from 10◦-65◦.)

Motivated by these experiments, we choose γ < 0 for

simplicity, which corresponds to a tilt angle of 90◦. In

principle, it is possible to modify the coupling term in

Eq. (3) such that nand Mare at an arbitrary angle,

but this makes the expression considerably more compli-

cated. The emergence of biaxiality (or the presence of

two distinguished directions) in the NLCs for non-zero

values of γ < 0 can be understood from the schematic

in Fig. 1: Choosing Malong the positive x-axis, the

LC molecules can align in two orthogonal directions, say

along the y-axis and z-axis.

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EmergenceofBiaxialityinNematicLiquidCrystalswithMagneticInclusions:SomeTheoreticalInsightsAdityaVats,1SanjayPuri,2andVarshaBanerjee11DepartmentofPhysics,IndianInstituteofTechnologyDelhi,NewDelhi{110016,India.2SchoolofPhysicalSciences,JawaharlalNehruUniversity,NewDelhi{110067,India.Thebiaxialphaseinn...

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Emergence of Biaxiality in Nematic Liquid Crystals with Magnetic Inclusions Some Theoretical Insights Aditya Vats1Sanjay Puri2and Varsha Banerjee1

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