PHYSICAL REVIEW E 94, 032701 (2016)

Dynamics of electro-orientation of birefringent microsheets in isotropic and nematic liquid crystals
M. V. Rasna,1 K. P. Zuhail,1 U. V. Ramudu,2 R. Chandrasekar,2 and Surajit Dhara1,*
2

1
School of Physics, University of Hyderabad, Hyderabad 500046, India
School of Chemistry, University of Hyderabad, Hyderabad 500046, India(Received 11 July 2016; published 29 September 2016)

We study the dynamics of electric field driven multiaxis electro-orientation of birefringent microsheets in boththe isotropic and nematic phases of a liquid crystal. For a fixed direction of applied field in the isotropic phase,
there are two critical fields above which the microsheets show two orientations. In the nematic phase, it shows
three rotations in both planar and homeotropic cells. These orientations are observed at varying voltages andwide time scales and are explained based on the competing effect of the electric, elastic, and viscous torques.
The control of the orientation of anisotropic microparticles (both optically and geometrically) by transducing
external energy may be useful in electro-optics and photonics.
DOI: 10.1103/PhysRevE.94.032701
I. INTRODUCTION

Studies on colloidal self-assembly have been one of the
major driving themes in material science to obtain new
functional and advanced materials. For example, making
metamaterials requires control of the self-assembly of complex
anisotropic colloidal particles such as toroids, coupled rods,
and split ring resonators [1,2]. In this context, nematic liquid
crystals (NLCs) have proven to be a potential medium forachieving predesigned, self-assembled structures. In NLCsa rich variety of structures are obtained by exploiting thelong-range elastic interactions among dispersed colloids andthe topological defects associated with them [3–7]. It has beensuggested that tunability in such materials can be envisaged byexternal electric or magnetic fields [8–11]. In addition, there isa growing interest in studying the dynamics of the colloids inliquid crystals driven by external fields [12,13]. The collective
response of these colloids may lead to new engineered materials and requires a proper understanding of their electrokinetics.
Recently, in NLCs, several interesting electrokinetic responsesof the colloids have been reported [11–15]. The majority of
research on electric field driven dynamics in NLCs has reported
on spherical colloids having isotropic optical and dielectric
properties [8–11,14,15]. However, there are a few reports on
the dynamics of nonspherical colloids in NLCs [11,14,16–18].
The dynamics of platelet-type colloids having optical anddielectric anisotropies is expected to be interesting. In thispaper, we report the electric field driven multiaxis orientationof birefringent (single crystal) organic microsheets in both theisotropic and nematic phases of a liquid crystal. The electricfield driven dynamics of single crystal microsheets promise
potential applications in photonics and electro-optics.
II. EXPERIMENT

Rectangular microsheets were prepared as per the reported
procedure [19]. We used 5,10,15,20-tetra(p-tolyl)porphyrin asa building block to prepare nearly monodispersed microsheets[see Fig. 1(a)]. Microsheets are crystallized in tetrahydrofuran/water solvents. Typical lengths and widths are in the*sdsp@uohyd.ernet.in

2470-0045/2016/94(3)/032701(5)range of 9–10 μm and 4–5 μm, respectively. The average
thickness of the microsheet is about 500 nm. A single crystal
x-ray diffraction (XRD) study reveals the monoclinic structureof the crystals [19]. They are optically birefringent andthe birefringence (n) of the microsheets is about 0.25 [5].
The orientation, interaction, and laser assisted self-assembly ofthe microsheets in a nematic liquid crystal have been reported
by us recently [5].
The experimental cells with various spacings are fabricatedby using indium-tin-oxide coated glass plates that are spincoated with polyimide AL-1254 and cured at 180 ◦ C for 1 h.
They were rubbed in the antiparallel direction for homogeneous alignment of the nematic director. For homeotropicalignment, we used JALS-204 and cured at 200 ◦ C for 1 h.
Sinusoidal voltage at 1 kHz is applied across the cell using
a function generator (Tektronix-AFG 3102) and a voltage
amplifier (TEGAM-2350), and the field direction is always
perpendicular to the substrates. High-speed videomicroscopy
is used for studying the rotational dynamics of isolated
microsheets. We used pentyl cyano biphenyl (5CB), a positive dielectric anisotropy nematic liquid crystal ( = 8.2),
and MLC-6608, a nematic mixture with negative dielectricanisotropy ( = −4.2) in the present study.
III. RESULTS AND DISCUSSIONFirst, we present the field response of a microsheet in theisotropic phase of a 5CB liquid crystal. Figure 1(b) shows
the schematic view of a microsheet with corresponding bodyaxes along the length, breadth, and thickness. An isolated
microsheet is selected, and the field-dependent response isrecorded by the video camera (see S4:Video-1) [20]. Some representative images taken using polarizing optical microscopeat various electric fields are shown in Figs. 1(c)–1(e). Usually,in the isotropic phase of the bulk sample, the microsheets
are randomly oriented. However, in the cell when the cell
thickness is not too large, most of the microsheets are parallelto the cell substrates [Fig. 1(c)]. When the applied field isin the range E1 (= 0.08 V/μm) < E < E2 (= 0.4 V/μm), themicrosheets first rotate about the x axis [Fig. 1(d)], and at
higher fields, i.e., E > E2 (=0.4 V/μm), they rotate about thez axis [Fig. 1(e)]. These rotations are observed over a wide
range of frequencies (100 Hz to 1 MHz) of the applied electric

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FIG. 1. (a) Optical microscopy image of the rectangular microsheets. The inset shows the chemical structure of the compound.
(b) Schematic representation of the microsheet, with corresponding
body axes x, y, and z along the length, breadth, and thickness,
respectively. (c)–(e) Electric-field-dependent multiaxis orientation ofa microsheet in the isotropic phase of 5CB (see S4:Video-1 [20]).
Sequence of polarizing optical microscopy images taken at different
applied fields, (c) E = 0 V/μm, (d) E1 < E < E2 , (e) E > E2 . The
green arrow indicates the direction of the applied field (out of plane).

field. However. all the experiments in this study are done at a
fixed frequency of 1 kHz that is much above the ionic charge
relaxation frequency of 5CB liquid crystals. In the first rotation
the electric-field-induced torque along the x is greater than thatof the y axis (i.e., τx > τy ). In the second rotation the torquealong the z axis is greater than that of the y axis (i.e., τz > τy ).
Thus the orientation of microsheets in the isotropic phase canbe controlled by the application of an external electric field.
To study the rotational dynamics, we applied a field,
0.75 V/μm(>E2 ), and observed two rotations around thex and y axes with time (see Fig. 2; S5:Video-2 [20]).
The rotational dynamics of the dielectric particles under theaction of electric field in isotropic liquids has been studied
both experimentally and theoretically [1,21–26]. In thosereferences, various types of microparticles are modeled as
dielectric ellipsoids, and the dynamics is understood basedon the competing effect of the electric-field-induced torqueand hydrodynamic torque [21,24]. Here, we briefly mention
the relevant torques acting on the particles. Total torqueacting on a microsheet is given by τ = τe + τv , where τe =
 and P = λE gives the dipole moment, where
1/2 Re[P ×E]
λ is the polarizability. The viscous torque can be written
as τv = ζ dα/dt, where ζ is the rotation constant, which
depends on the medium viscosity, rotational hydrodynamic
drag coefficient, and particle geometry [20]. The rotation
angle about the x axis, namely, α [the angle between they axis and field direction as defined in Fig. 2(d)] decreases
from 90◦ to 0◦ [see S1(a) [20]]. The x component of theelectric torque is given by τe = 12 λ1 E 2 cos α sin α. This isbalanced by the viscous torque, and the corresponding rotation
rate can be describe by tan α = tan α0 exp(−t/Tα ), where thetime constant is given by Tα = ζ /(E 2 λ1 ). The polarizability
anisotropy is given by λ1 = (λy − λz ), where λy and λz are
the polarizabilities of the ellipsoid along the y and z directions[20]. Figure 2(g) shows a typical variation of tan α with time.

PHYSICAL REVIEW E 94, 032701 (2016)

FIG. 2. (a)–(c) Time-dependent multiaxis rotation of a microsheet in the isotropic phase of 5CB for a fixed applied field,
0.75 V/μm(>E2 ) (see S5:Video-2 [20]). (d)–(f) Schematic representations of corresponding orientations. The green arrow indicatesthe direction of field. α defines the angle between the y axis and theelectric field. φ defines the angle between the x axis and E. Variationof (g) tan α and (h) tan φ with time. Red curves show the best fit to
the experimental data. Insets: Variation of Tα and Tφ with 1/E 2 . The
red lines are the best fits.

This experiment was performed at various fields, and the time
constant of the decay (Tα ) is found to be linearly proportional
to 1/E 2 [the inset of Fig. 2(g)], and the slope of the linear fit isgiven by ζ /λ1 = 0.0071 s(V/μm)−2 . After completing therotation about the x axis, the microsheet rotates about the zaxis and this rotation is slower than the rotation along the xaxis. In the second rotation, φ [the angle between the x axis andfield direction as defined in Fig. 2(e)] decreases from 90◦ to 0◦[see S1(b) [20]], and the rate of rotation can again be describedby tan φ = tan φ0 exp(−t/Tφ ), where Tφ = ζ /(E 2 λ2 ) andλ2 = λx − λy [20]. Figure 2(h) shows a typical variation oftan φ with time. This experiment was also repeated at various
fields and the time constant of the decay (Tφ ) is found to be
proportional to 1/E 2 [see the inset of Fig. 2(h)]. The slope ofthe linear fit is given by ζ /λ2 = 0.1469 s(V/μm)−2 . In theisotropic phase two independent rotations around two mutuallyorthogonal axes suggest that the microsheets are dielectricallyanisotropic. In this context it may be recalled that from the
x-ray diffraction studies it was found that the particles have amonoclinic crystal structure [19].
In the nematic phase, first we discuss the rotation of the
microsheets in the planar cell below the Freedericksz thresholdfield (see S6:Video-3) [20]. The microsheet does not respondto the field below a critical value (E3 ) and the plane of the
microsheet is parallel to the substrate [Fig. 3(a)]. There is anintermediate and narrow field range, E3 (0.09 V/μm) < E <Ef (= 0.11 V/μm) (Ef is the Freedericksz threshold field), in

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PHYSICAL REVIEW E 94, 032701 (2016)

FIG. 3. Time-dependent rotation of a microsheet in nematic
planar cell (5CB) below the Freedericksz threshold field. (a), (b)Sequence of images for the applied field E = 0.1 V/μm in 5CB at
different times, where E3 (= 0.09 V/μm) < E < Ef (= 0.11 V/μm).
Schematic top view of the microsheet for (c) E < E3 and (d)E3 < E < Ef . α defines the angle between the y axis and field
direction. (e) Variation of tan α as a function of time, and the redline shows the best fit (see S6:Video-3 [20]). n represents director(the average alignment direction of the LC molecules).

which the microsheet rotates about the x axis, and the angle
α [the angle between the y axis and field direction as definedin Fig. 3(c)] decreases from 90◦ to 0◦ [Figs. 3(a) and 3(b); seealso S2 [20]). This rotation is due to the electric-field-induced
torque and is similar to that observed in the isotropic phase[Figs. 2(a) and 2(b)], but is somewhat slower compared tothe isotropic phase. In particular, in the isotropic phase, therotation was completed in less than 0.1 s [Fig. 2(g)], whereas in
the nematic phase it took about 0.24 s [Fig. 3(e)]. This rotationis also due to the competing effects of viscous and electric
torques, as discussed previously. Hence it is legitimate to use
the same model and neglect the effect of small elastic torqueexperienced by the microsheets while rotating. Figure 3(e)shows the time variation of tan α and the corresponding fit totan α = tan α0 exp(−t/Tα ). The time constant obtained, Tα =100 ms, is much larger than that of the isotropic phase [seeFig. 2(g)]. This is expected as Tα is directly proportional to theaverage viscosity of the medium.
Now we discuss the rotation of the microsheets at afield which is greater that the Freedericksz threshold field(Ef = 0.11 V/μm) (see S8:Video-4 [20]). Figure 4 shows
the rotation of the microsheet at an applied field, E =0.4 V/μm. Since E > Ef , the nematic director reorientsalong the field direction and consequently the long axis of
the microsheet gets oriented, followed by an in-plane tilt
[the third image in Fig. 4(a)]. These rotations are somewhat
similar to those reported in square-type platelets with ahole in the middle (made of SU-8 photoresist) [11], except
the final orientation of the microsheet is very different in
the present case. Figure 4(e) shows the variation of φ(t)[the angle between the electric field and the body axis x
as defined in Fig. 4(b)] for two different field amplitudes.
It can be fitted to a Boltzmann function φ(t) = (φ2 − φ1 )/{1 + exp[(t − t◦ )/Tφ ]} + φ2 , where (φ2 − φ1 ) is the totalchange of φ, t◦ is the time at which φ(t) = (φ2 − φ1 )/2, andTφ is the relaxation time. Figure 4(f) shows the relaxation timeTφ as a function of field. Tφ is expected to follow the following
field dependence [11], Tφ = (γ d 2 /π 2 K)[(E/Ef )2 − 1]−1 ,FIG. 4. Rotation of a microsheet in nematic planar cell (5CB) ata fixed field (E = 0.4 V/μm) (see S7:Video-4 [20]). (a) Sequenceof images of the microsheet taken at different times. (b)–(d)Corresponding schematic top view. (e) Variation of φ(t) for two
different field amplitudes, and the solid red line shows the best fits tothe data with the Boltzmann function. (f) Relaxation time Tφ , obtained
from the fitting (circles) and calculated (red line), as a function offield. (g) Variation of in-plane rotation θ(t) for E = 0.25 V/μm and
0.4 V/μm. (h) Variation of 1/Tθ with E. Red lines show the best fits
to the experimental data. n indicates the director.

where γ is the rotational viscosity, d the cell thickness, andK the average Frank elastic constant. Taking typical values
γ = 87 mPa s, K = 6.5 pN, and Ef = 0.11 V/μm for 5CB,
the calculated Tφ [solid red lines in Fig. 4(f)] agrees wellwith the experimental results. The in-plane rotation θ [see thethird image in Fig. 4(a)] arises from the elastic interactionof the particle-induced distortions of the director and thefield-dependent director deformations near the surface. For
E > Ef the splay bend deformation occurs over a heighth ≈ (d/E) oK ≈ d(Ef /Eπ ), where K is the average elastic constant and  is the dielectric anisotropy. The elastic freeenergy is given by [11] Fel ≈ K(ac/ξ )(θ − θ∞ )2 /2, wherea and c are the average length and thickness of the sheets.
The elastic torque τel = −∂Fel /∂θ ≈ −K(θ − θ∞ )ac/ h isbalanced by a viscous drag torque τv = ξ ηdθ/dt, where ξ isthe rotational drag coefficient. The solutions of the equationsof motion are exponential, θ (t) − θ0 = θ1 exp(−t/Tθ ), whereθ0 is the equilibrium angle and Tθ is the time constant,

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PHYSICAL REVIEW E 94, 032701 (2016)

1/Tθ = π Kac/(D 3 ηd)(E/Ef ), where d is the cell thickness
and D the diagonal length [11]. Figure 4(h) shows a linear
variation of 1/Tθ with E, and the corresponding slope isgiven by 1.9 (sV/μm)−1 . Taking typical values K = 6.5 pN
and η = 25 mPa s and other geometrical parameters of the
microsheets, the calculated slope is 1.3 (sV/μm)−1 , which
is very close to the experimental value despite the number ofapproximations made in the calculation.
The microsheet can be orientated vertically in ahomeotropic cell. In this case a nematic liquid crystal withnegative dielectric anisotropy is needed to study the electricfield driven dynamics. For this purpose we have chosen anematic mixture commercially known as MLC-6608. To study
the dynamics we applied E = 0.4 V/μm, which is greaterthan the Freedericksz threshold field (Ef = 0.16 V/μm), andrecorded the time-dependent rotations. The director reorientsabove the threshold field and consequently the microsheet
rotates about the y axis, followed by an in-plane rotation

about the z axis (see S8:Video-5) [20]. Figure 5(a) showsa series of images of rotation and Figs. 5(b)–5(d) show thecorresponding schematic top views. After the in-plane rotation,
it further rotates about the x axis, making the y axis parallelto the direction of the applied field. Figure 5(e) shows theplot of angle φ(t) (the angle between the direction of electricfield and the z axis of the microsheet) for two different field
amplitudes. These can also be fitted with the Boltzmann
function φ(t) = (φ2 − φ1 )/{1 + exp[(t − t◦ )/Tφ ]} + φ2 , andthe corresponding relaxation times Tφ at various fields areshown in Fig. 5(f). Taking typical values γ = 0.186 Pa s, K =17.4 pN, and Ef = 0.16 V/μm, for MLC-6608, the calculated
Tφ at room temperature agrees well with the experimental
values. Figure 5(g) shows the time variation of the in-planerotation angle θ . It is observed that θ increases from zero andsaturates to about 33◦ within 1 s. The rotation about the x axis isdue to the electric-field-induced torque, and the corresponding
angle of rotation is defined as α [the angle between the y axis
and the field direction as defined in Fig. 5(c)]. The variationof α with time is given in the Supplemental Material [20].
Figure 5(h) shows the time variation of tan α together withthe best fit to tan α = tan α0 exp(−t/Tα ). The time constant(Tα ) varies linearly with 1/E 2 [see the inset for Fig. 5(h)].
The slope of the linear fit is given by 0.015 s(V/μm)2 . The
ratio of the slopes obtained from the linear fit of Tα vs 1/E 2 in
the isotropic phase of 5CB [see the inset of Fig. 2(g)] and thenematic phase of MLC-6608 is 0.46, and this is comparable
MLC-6608
5CB
to the ratio of the average viscosities (ηiso/ηN= 0.5)of the respective phases. A notable difference between the
final orientation (above the Freedericksz threshold field) in thenematic phase of 5CB and MLC-6608 may be mentioned. In
5CB, the x axis of the microsheet is oriented parallel to thefield, whereas in MLC-6608 the y axis is orientated along thefield direction.
IV. CONCLUSION

FIG. 5. Rotation of the microsheet in a homeotropic cellof MLC-6608 liquid crystal under the action of a fixed field,
E = 0.4 V/μm (>Ef ) (see S8:Video-5 [20]). (a) Sequence of images
taken at different times and (b)–(d) corresponding schematic top
views. (e)–(g) Time variation of rotation angles φ and θ for two
different field amplitudes; the red curve in (e) shows the best fit
to the Boltzmann function. (f) Variation of experimental relaxation
time Tφ with field, together with the calculated values (red lines)
for MLC-6608. (h) Variation of tan α with time; red lines give theexponential fitting. Inset: Linear variation of Tα with 1/E 2 . n indicatesthe director.

In conclusion, we have shown that by applying a unidirectional electric field, birefringent microsheets can be rotatedabout multiple axes both in the isotropic and nematic phases ofliquid crystals. Both positive and negative dielectric anisotropy
liquid crystals are studied in the experiment. Two rotations in
the isotropic phase are due to the electric-field-induced torques
and suggest that the microsheets are dielectrically biaxial. In
nematic planar and in homeotropic cells a total of six rotations
are observed due to the interplay between electric, elastic, andhydrodynamic torques of the liquid crystal medium. The rateof rotation depends on the amplitude of the applied field andthe average viscosity of the medium. A liquid crystal mediatedrearrangement of optically anisotropic colloids may be potentially useful in applications in the field of photonics, for developing reconfigurable composites for metamaterials, and forunderstanding active matter. The analysis may also be helpful
for the biophysical studies of anisotropic biological particles.
ACKNOWLEDGMENTS

We gratefully acknowledge the support from the DST(DST/SJF/PSA-02/2014-2015), and DST-PURSE. M.V.R. acknowledge UGC-BSR for support through a fellowship grant.

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