PHYSICAL REVIEW E 90, 012506 (2014)

Rheology of twist-grain-boundary- A liquid crystals
Rasmita Sahoo,1 J. Ananthaiah,1 R. Dabrowski,2 and Surajit Dhara1,*
1
School of Physics, University of Hyderabad, Hyderabad 500046, India
Institute of Chemistry, Military University of Technology, 00-908, Warsaw, Poland(Received 20 January 2014; revised manuscript received 14 April 2014; published 11 July 2014)2

We report studies on the rheological properties of a liquid crystalline analog of Abrikosov phase in type-II
superconductors known as twist-grain-boundary-A (TGBA ) phase. The TGBA phase shows a large apparent yield
stress compared to the cholesteric (N∗ ) phase. The storage modulus (G ) of the TGBA phase is significantlylarger than the loss modulus (G ). The dynamic relaxation measurements indicate a solid-like behavior of N∗ ,TGBA , and smectic-C ∗ phases. The complex shear modulus of the TGBA phase exhibits a power-law behaviorG∗ (ω) ∼ ωα with α  0.5. The relative amplitude of G and G at various temperatures indicate that the enhanced
elasticity of TGBA phase is due to the structural defects.
DOI: 10.1103/PhysRevE.90.012506

PACS number(s): 83.80.Xz, 83.60.La

I. INTRODUCTION

phase, well-defined SmA blocks are separated by regularly
spaced planar arrays of parallel screw dislocations (Fig. 1). The
normal layer of the successive slabs rotate relative to each otherby an angle. The contrary structure of smectic-like layeringand cholesteric-like twist was experimentally discovered inhighly chiral liquid crystals by Goodby et al. [9] and wascharacterized by several experiments [8,10]. Soon after the
discovery there were some reports on the observation of several
variants of TGBA , such as a TGBC , TGBC ∗ [10–12]. In abinary mixture of chiral and nonchiral compounds, Pramod
et al. [12] reported a three-dimensionally modulated chiralsmectic-C phase called UTGBC ∗ which has a two-dimensional(2D) undulation of SmC ∗ blocks in the form of a squarelattice. There are several experimental reports on the structure,electro-optic, and dielectric properties of various TGB liquid
crystals [13,14]. However, the viscoelastic properties of thesephases have remained unexplored. In this paper we reportrheological measurements on a binary mixture exhibitinga large temperature range (18 ◦ C) of the TGBA phase.
The experimental results are analyzed based on the known
theoretical model.

Structural disorder or defect in general affects the mechanical properties of all materials. In solid materials the disorder
or the defects are created or pinned down by impurities fordesired mechanical properties. Many anisotropic fluids suchas liquid crystals possess topological defects. The networkof defects in such systems can respond to shear, and hence
their contribution to the rheological properties is significant.
The density of defects in liquid crystals can be enhanced byadding colloidal particles in cholesteric (N∗ ) liquid crystals,
aerosil in smectic-A (SmA) liquid crystals [1–3]. Recently
the enhancement of elasticity due to the network of defectlines entangled with colloidal particles in liquid crystals has
been reported [4]. Generally, the viscoelastic properties insuch composite systems are enhanced and their dynamics canbe explained based on the soft-glassy-rheology or the theoryof rubber elasticity. However, soft materials with long-range
periodic structural disorder due to the defects (dislocations)and the grain boundary composed of an array of dislocations
are rare and their viscoelastic properties are still unexplored.
One such very unusual soft material is type-II smectic-A,
known as the twist-grain-boundary-A (TGBA ) liquid crystal.
The TGBA is a liquid crystalline analog of the Abrikosov
phase exhibited by type-II superconductors [5]. The analogy
between superconductors and smectic liquid crystals was
proposed by de Gennes, who predicted the intermediatephase with a lattice of dislocations in smectics. Renn andLubensky showed that the intermediate phase of highly
chiral type-II materials consists of a twisted arrangementof blocks of SmA liquid crystals, separated by twist grainboundaries, which are made of an array of screw dislocations[6,7]. A typical structure of the TGBA phase is shown in
Fig. 1.
The interacting dislocations form the grain boundary andthe interaction among the grain boundaries stabilizes the TGBA
structure. A typical distance between two dislocations (ld ) ina grain boundary and the intergrain boundary distance (lb )is in the range of ld ∼ lb ∼ 20 to 30 nm [8]. The smectic-A
blocks with these dislocations form a helical structure. In this*

Corresponding author: sdsp@uohyd.ernet.in

1539-3755/2014/90(1)/012506(5)II. EXPERIMENT

We prepared a binary mixture of two compounds namely,
4-(2 -methyl butyl phenyl 4 -n-octylbiphenyl-4-carboxylate(CE8) and 2-cyano-4-heptyl-phenyl-4 -pentyl-4-biphenyl carboxylate [7(CN)5]. CE8 is a chiral and 7(CN)5 is a nonchiralcompound and both were synthesized in our laboratory. The
detailed phase diagram of the binary mixtures was reportedby Pramod et al. [12]. Based on the phase diagram, weprepared a mixture of 63.6 wt% of CE8 and 36.4 wt% of7(CN)5 to get a large temperature range of the TGBA phase.
The mixture was prepared by weighing each component ina glass container. The sample was then heated above theisotropic phase and mixed physically by using a thin glassrod. Polarized optical microscope shows following phasetransitions on cooling: I 120.2◦ C BP 119◦ C N∗ 80◦ C TGBA
62.1◦ C UTGBC ∗ 59.8◦ C SmC ∗ . It may be noted that thetemperature range of TGBA is about 18◦ C and UTGBC ∗ is very
short, about 2◦ C. The temperature range of TGBA is muchlarger than that usually observed in a single liquid crystal
compound. Rheological measurements were made by using

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PHYSICAL REVIEW E 90, 012506 (2014)

FIG. 1. (Color online) Schematic representation of structure of
the TGBA phase. The distances between two dislocations and two
grain boundaries are ld and lb , respectively.

a stress-controlled rheometer (Anton Paar MCR 501) withthe cone and plate measuring system of diameter 25 mm,cone angle 1◦ , and with a minimum gap of 50 μm (the gap
between the plate and the truncated cone). The temperatureof the sample was controlled with an accuracy of 0.1◦ C
by a Peltier temperature controller fitted with a hood forthe uniformity of the sample temperature. The calibration of
the temperature controller was checked by measuring the phasetransition temperatures of some standard liquid crystalline
materials [15–18]. The rheological properties of many soft
materials, especially liquid crystals, when freshly loaded intoa rheometer are usually not reproducible because a large
degree of macroscopic disorder and inhomogeneity that may
get introduced during the loading process [19]. Hence weheated the sample to the isotropic phase and cooled down to
the liquid crystalline phase. The sample is then presheared
for about 400 s with a shear-rate of 10 s−1 before themeasurements.
III. RESULTS AND DISCUSSION
A. Texture and physical observations

The typical textures of various phases observed under
optical polarizing microscope in a planar cell (a cell treated forplanar alignment of the director, i.e., the average direction ofmolecular alignment) is shown in Fig. 2. The N∗ phase exhibits

FIG. 2. (Color online) Textures observed in polarizing optical
microscope (POM) at different temperatures (a) 110◦ C (N∗ ) (b) 73◦ C(TGBA ), and (c) 55◦ C (SmC ∗ ). A square grid pattern characteristics
of UTGBC ∗ appears between TGBA and SmC ∗ is also shown in thebox. The physical appearance of the sample while trying to pull it up
by a glass rod at corresponding temperatures are also shown.

the typical defect network of the “oily-streak” [Fig. 2(a)]. The
TGBA phase shows almost homogenous texture and that theslight nonuniformity in the color is due to the nonuniformalignment of the helix axis at different regions ([Fig. 2(b)].
There is also a short temperature range of the UTGBC ∗phase below the TGBA phase with a characteristic square-gridpattern. As the temperature is lowered, SmC ∗ appears to be
keeping to the square grid pattern [Fig. 2(c)], and it can beremoved by slightly shearing one of the plates. The physicalappearance of the sample under mechanical disturbance suchas pulling up by a glass rod at various temperatures (after
mounting it on the rheometer) is also shown below therespective textures in Fig. 2. It appears that N∗ tends to flow
easily like a liquid. The TGBA phase appears like a sticky fluidand SmC ∗ is kind of a semisolid that does not flow easily.
B. Yield-stress measurements

The determination of true yield stress is rather difficult andit depends on the measuring techniques [20–22]. However,
the apparent yield stress can be estimated from the shear-ratedependent shear-stress for relative comparison in the different
phases of the same sample. Figure 3 shows the shear-ratedependent effective shear stress (σeff ) at different temperatures.
At the low shear rate (γ̇ < 0.1 s−1 ), the stress tends to reachalmost a constant value as γ̇ −→ 0, showing an apparent yieldstress (σy ). At 110◦ C (N∗ ), σy is negligibly small (  10−2 Pa)
and it increases as the temperature is lowered in the TGBA andSmC ∗ phases. For example, at 68◦ C (TGBA ) σy 10 Pa andit further increases as the temperature is decreased to SmC ∗(57◦ C) phase. Very small σy in the N∗ phase is due to theresidual oily-streak defect network of the presheared sample.
Thus, σy is three orders of magnitude larger in TGBA thanthat of N∗ phase. The apparent yield stresses have also beenreported in many other smectic materials [23,24]. However,
the apparent yield stress in TGBA and in SmC ∗ phases of thepresent system is much larger than that known in low molecularweight smectic liquid crystals [23,25]. The N∗ phase shows
shear thinning followed by a Newtonian behavior at highshear rates (>10 s−1 ). In the intermediate shear-rate range,i.e., 0.1 < γ̇ < 100 s−1 , it shows shear thinning behavior

FIG. 3. (Color online) Shear-rate-dependent shear stress (σeff )at different temperatures N∗ : [110◦ C (squares)], near N∗ -TGBA
transition: [81◦ C (circles)], TGBA : [73◦ C (uptriangles)], TGBA :[68◦ C (downtriangles)], SmC ∗ : [57◦ C (diamonds)].

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in all phases. The shear-rate-dependent viscosity of nematicor cholesteric liquid crystals is usually characterized by the
Ericksen number [26], which is given by the ratio of theflow-induced viscous stress (ηγ̇ ) to the Frank stress (K/ h2 ),where K is the average curvature elastic constant and h is thelength scale of the flow geometry. Theoretically it has beenshown that in the intermediate range of the Ericksen number(10 to 400), the shear viscosity shows a shear thinning behavior[27]. Assuming a typical average value of average curvatureelastic constant K ∼ 10−11 N and measured apparent viscosityof the N∗ phase η  10 mPa, the estimated Ericksen numberis 300, which is in the theoretically predicted range. Atvery high shear rate (γ̇ > 100 s−1 ), σeff in both the TGBA
and SmC ∗ phases fall suddenly to a much lower value. Thissudden decrease could be connected to the wall slip or theplastic deformation of the material.
C. Dynamic relaxation measurements

To determine the linear regime of viscoelasticity by
oscillatory measurement, we measured the strain amplitude
dependence of storage (G ) and loss (G ) moduli at various
temperatures. The measurement is performed every time ona fresh sample and the results are shown in Fig. 4. The datain the N∗ phase (110◦ C) at low strain amplitude are noisy
and are attributed to the lower limit that our rheometer can
reproducibly measure. Nevertheless, the average values ofboth G and G are independent of strain up to γ  0.8 inall phases. In the TGBA phase, (73 and 68◦ C) in the linearviscoelastic region G > G and G of TGBA is more than
1000 times larger than that of the N∗ phase. Similarly, inthe SmC ∗ phase (57◦ C), G > G and G is about 2.5 times
larger than that of the TGBA phase. The viscoelasticity of
the N∗ phase is mostly contributed by the three-dimensional
network of residual oily-streak defects [1]. The SmA liquid
crystals will have a larger viscoelastic response than nematicor N∗ merely because of the positional order. For example, in
our previous studies, in cholesterol nonanoate, (exhibits N*to the SmA transition) we found GSmA ≈ 80 GN ∗ [28]. In

FIG. 4. (Color online) The strain amplitude dependence of the
storage G (solid symbols) and the loss G (open symbols) moduli at◦different temperatures. 110 C (hexagons) [N∗ ], 73◦ C (diamonds)[TGBA ], 68◦ C (circles) [TGBA ], and 59◦ C (squares) [SmC ∗ ] at
ω = 1 rad/s.



the present sample, we find GTGBA ≈ 1000 GN∗ . Hence, theenhanced elasticity of the TGBA phase can be attributed tothe bulk sample with structural defects present in the system.
In this phase the enhancement in the viscoelasticity may
also be due to the energy cost for the deformation of grainboundaries. In addition, the surface tension of the smectic
layers which acts against the deformation plays a vital rolefor increasing the shear modulus. In the SmC ∗ phase, there
is no grain boundary, however, the individual dislocations
and their motions contribute to the viscoelastic properties.
In fact, it has been suggested that the shear modulus isstrongly influenced by the line tension of screw dislocations
because it acts against the Peach-Koehler force which controls
the motion of the dislocations [29]. When the force exerted
on the dislocations dominates over the line tension, thenthe system overcomes its elastic deformation and undergoes
plastic deformation. The drastic fall of σeff shown in Fig. 3 forboth the TGBA and SmC ∗ phases could be due to the plastic
deformation.
The role of defects and disorder on the viscoelasticity of
randomly oriented type-I smectic liquid crystals has been
discussed by many authors [2,3,29–35]. In those experiments,
the random orientation and defects were either stabilized
by quench disorder or by putting colloidal particles in thesamples. Basappa et al. [3] showed in the lyotropic lamellar
system that adding a particle enhances the moduli and thedissipation and the elasticity arises mainly due to the defectnetwork. Bandyopadhyay et al. [2] showed that the elasticity
of 8CB (octyl cyanobiphenyl) liquid crystal in the SmA phase,
confined in the aerosil network, was enhanced compared tothe pristine sample. The shear modulus of such systems show
a power-law behaviorG (ω) = G0 + βωα ,(1)where G0 is the plateau modulus arising from the zerofrequency shear modulus. In the present sample, TGBA is
composed of rotating SmA blocks, and hence the randomorientation of layers is inherent to the structure, which cannot
be removed by the effect of shear. Thus the dynamic responseof our sample can be analyzed based on this model. In
Fig. 5 we show some representative frequency dependenceof storage and loss moduli in all phases. In the N∗ phase,
G > G , and at the low frequency region (ω  2 rad/s) bothare constant. They increase with frequency beyond this range.
For example, beyond ω > 2 rad/s they vary as G ∼ ω2 andG ∼ ω. It may be mentioned that solid-like behavior in the
low frequency region (ω = 1 rad/s) was also observed incolloidal-particle-dispersed cholesteric liquid crystals. But at
higher frequency the same sample showed fluid-like behavior(G < G ) [1].
The fit parameters at different temperatures are shown in
Fig. 6(a). In the N∗ phase (90 to 110◦ C), α obtained from both
the G and G data are almost close to 2 and 1, respectively.
It is interesting to note that the exponent α obtained fromboth the fittings of G and G decreases rapidly (below 90◦ C)
and is almost equal, i.e., α  0.5. Ramos et al. [1] showedthat the storage modulus of liquid crystalline systems witha defect network has contributions both from the disorientedpart of the sample [G (ω) ∝ ω1/2 ] and regions of the sample

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PHYSICAL REVIEW E 90, 012506 (2014)

FIG. 5. (Color online) The frequency dependence of (a) storageG (solid symbols) and (b) loss G (open symbols) moduli at three
representative temperatures in all the phases. The solid lines aretheoretical fits to Eq. (1).


where the layers are parallel to the shear direction [G (ω) ∝ω2 ]. Similarly, the loss modulus also has contribution from
disoriented parts of the sample (G (ω) ∝ ω1/2 ) and Maxwellfluid type a contribution [G (ω) ∝ ω]. Comparing to our
results, it appears that the storage and loss moduli of TGBA
has a significant contribution from the disoriented layers of
the sample only. The temperature dependence of Go and βobtained by fitting G (ω) and G (ω) is also shown in Fig. 6(a).
◦For example, at 110 C, Go obtained from both the fits
are 0.25 and 0.12, respectively. At the same temperature, βvalues are 0.01 and 0.03, respectively, and similar values werealso reported in other cholesteric samples [1]. The plateau
modulus usually arises from the elasticity of static defects
in lamellar systems [1,2]. In the present system Go is muchlarger (10 times) than the low molecular weight thermotropic
SmA liquid crystal, such as 8CB [2]. The variation of Go with
reduced temperature χ ≡ (TN∗ −TGBA − T )/TN∗ −TGBA is shownin Fig. 6(b). In the TGBA and SmC ∗ phases, Go is more than
three orders of magnitude larger than the N∗ phase [1]. In
analogy with rubber elasticity, the contribution to the elastic
response of a static defect network should vary as Go ≈ τ/d 2 ,where τ is the line tension and d is a typical average spacing
between defects [1]. In the case of screw dislocations thedefect line tension can be written as τ = Bb4 /128π 3 rc2 , whereb = mdo is the Burger’s vector of integer strength m and rc isthe defect core radius [36]. Measurements on many smectic
systems show that B ∼ χ 0.4 [37] and rc−2 ∼ ψ 2 ∼ χ 0.5 , whereψ is the smectic order parameter [38]. Thus τ ∼ χ 0.9 , in close
agreement with Go ∼ χ γ with γ ≈ 1 [Fig. 6(b)], suggestingthat the defects (screw dislocations) significantly contribute to
the shear response. It may also be noted that the exponent α
in the TGBA is 0.5, and further it tends to decrease in the

FIG. 6. (Color online) (a) The exponent α characterizing thepower-law contribution to G (ω)(solid squares) and G (ω) (opensquares) as a function of temperature. (Inset) The inset displays
the fit parameters Go and β obtained from G (ω) and G (ω) data.
Diamonds (solid and open) and circles (solid and open) are obtainedfrom G (ω) and G (ω) data, respectively. (b) Variation of Go with
reduced temperature χ . The solid line showing the best-fit result
Go ≈ χ γ , with γ = 1.

SmC ∗ phase. This is somewhat less than that of the lamellar
block copolymer α = 0.6 [39]. Within the soft glass rheology
model, the glass transition is characterized from the power-law

exponent G ∼ ωx−1 . In this model x, is an effective noise
temperature and in the range 1 < x < 2, G and G have aconstant ratio [19]. The system approaches the glass transition
as x −→ 1, i.e., α = (x − 1) −→ 0. In the N∗ , α obtainedfrom the fitting of G and G is very different and it decreases
with decreasing temperature and merges to a single value(0.5) upon entering the TGBA and SmC ∗ phases. Though
the TGBA and SmC ∗ are thermodynamically stable phases,
the experimental x value is about 1.5, indicating that theirrheological responses are similar to many soft glassy materials.
It may be mentioned that the adapted model does not include
the effect of defect nucleation and annihilation. Theoretically,shear-induced defect nucleation and annihilation in a flowaligned nematic liquid crystalline polymer has been reported
by Grecov et al. [40]. Experimentally, the effect of such a
process on the shear rheology of low molecular weight liquid
crystals is not well established. Nevertheless, this effect is
expected to be less important in the TGBA as this phase isstabilized by interacting dislocations and the dislocations are
inherent to the structure.
IV. CONCLUSION

In conclusion, we have studied the rheology of a binary
mixture showing a large temperature range of the TGBA phase.
All the phases in the mixture, i.e., N∗ , TGBA , and SmC ∗exhibit solid-like behavior at low frequencies and low strain

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amplitude. The elasticity of the TGBA phase is discussed
based on the theory in lamellar systems that accounts forthe contribution of defects. The power-law dependence of thecomplex shear modulus and the analysis of the experimental
result suggests that the elasticity of the TGBA phase is due tothe structural defects. A more quantitative assessment shouldbe able to provide the prominence of screw dislocations andperhaps the grain boundaries. The effective noise temperatureapproaches 1, suggesting TGBA liquid crystals and its low

temperature phases are a defect-mediated soft solid whose
dynamics is similar to soft glassy materials.

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ACKNOWLEDGMENTS

We gratefully acknowledge the support from the CSIR
(03(1207)/12/EMR-II. R.S. acknowledge UGC for BSR fellowship.

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