fluids
Review

A Short Review on the Rheology of Twist Grain
Boundary-A and Blue Phase Liquid Crystals
Rasmita Sahoo and Surajit Dhara *School of Physics, University of Hyderabad, Hyderabad 500046, India; rasmita.phy@gmail.com* Correspondence: sdsp@uohyd.ernet.in
Received: 22 February 2018; Accepted: 4 April 2018; Published: date


Abstract: Topological defects are important in determining the properties of physical systems andare known varyingly depending on the broken symmetry. In superfluid helium, they are called
vortices; in periodic crystals, one refers to dislocations; and in liquid crystals, they are disclinations.
The defects and the inter-defect interaction in some highly chiral liquid crystals stabilize someintermediate complex phases such as Blue Phases (BPs) and Twist Grain Boundary-A (TGBA ) phases.
The defect dynamics of these phases contributes to the rheological properties. The temperature rangeof these intermediate phases usually are very small in pure liquid crystals; consequently, a detailed
experiment has been difficult to achieve. However, the temperature range could be enhanced
significantly in multicomponent systems. In this review article, we discuss some recent experimental
progress made in understanding the rheological properties of the wide-temperature-range TGBA andBP liquid crystals.
Keywords: liquid crystals; topological defects; TGB A , BPs; rheology

1. Introduction
Liquid crystals belong to the broad class of soft matter, which appears in between crystalline
solids and isotropic liquids. They are usually made of shape anisotropic organic molecules andexhibit a variety of phase transitions. A nematic is one of the simplest liquid crystals in which
the constituent molecules have a preference to align with one another along one orientation in thesystem called the director and denoted by a dimensionless unit vector n̂. The nematic is apolar, i.e.,n̂ and −n̂ are physically equivalent. In the nematic phase, two different types of topological defects
or singularities are observed. These are either line defects, known as disclinations (showing thediscontinuity in the inclination of molecules) or point defects (hedgehogs and boojums) where thedirector cannot be defined [1,2]. These defects cannot be eliminated by continuous deformation of theorder parameter field; hence, they are topologically stable [3]. They are characterized by strength orwinding number ‘s’, which is denoted by an integer or half-integer number. The strength ‘s’ determines
the number of times the director rotates around the singularity when one encircles the defect core [2].
Two oppositely-charged defects can annihilate one other. The orientation of the nematic director field(ψ) surrounding the defect can be expressed in terms of defect strength (s) [1]:
ψ = sα + β(1)where α = tan−1 (y/x ) and β is a constant. Here, the defect strength s depends on the symmetry of themedium. In the nematic phase, s is the multiple of ± 12 , as the director has n̂ ≡ −n̂ symmetry. Figure 1
shows the schematic representation of the director structure of a few defects.

Fluids 2018, 3, 26; doi:10.3390/fluids3020026

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Figure 1. Schematic representation of a few defects with different winding numbers. Continuous lines
represent the director field around the singularity.

In the cholesteric phase (N* ), the molecular chirality induces spontaneous twist, which leads tocomplicated, but stable networks of disclination lines in a planar cell, known as oily-streak defects [4].
These defects are easily observed under a polarizing optical microscope (POM). Smectic liquid crystals
exhibit dislocations due to the layered structure. In this case, the distortion alters the smectic layer
displacement field u, which leads to the dislocations in the medium and is defined as [3]:IΓ

du = (kd)ẑ = b(2)

where ẑ signifies the direction of the lattice distortion, d is the layer spacing and b is called Burger’s
vector. Here, the nature of the defects depends on the relative orientations of b and the dislocation
line. A schematic representation of dislocations in the SmA phase is shown in Figure 2. If b is parallelto the dislocation line, it is called the “screw dislocations” (Figure 2a). The smectic layers climb up
or down by an integral number of d upon going round the dislocation line, changing in a screw-likemanner. On the other hand, if b is perpendicular to the dislocation line, it is called an “edge dislocation”(Figure 2b). In this case, a few layers are either added or removed from one side of the dislocation
line. The dislocation lines cannot be observed under POM. In addition to dislocations, smectic liquid
crystals also show focal conic defects, and these defects can be observed under POM [2].

(a)

(b)

Figure 2. (Color online) (a) Schematic representation of screw dislocations in the SmA phase.
The cross-section shown here indicates that each layer changes height by half a layer in half a circuit ofthe defect core. The dislocation line is denoted by I, parallel to Burger’s vector and b = +d. (b) An edge
dislocation with b = −d.

1.1. Twist Grain Boundary Phase
In highly chiral liquid crystals in between the cholesteric and SmA phase, occasionally,an intermediate phase with a relatively larger pitch is observed. In smectics, when molecular chirality

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increases, the competition between the twist formation and smectic layer formation leads to somefrustrated structures, known as twist grain boundary phases [5]. This is a thermodynamically-stablephase. Experimentally, it was first observed by Goodby et al. [6]. The Twist Grain Boundary-A(TGBA ) phase has an analogy with the Abrikosov phase of Type-II superconductors in an external
magnetic field, and it was first proposed by de Gennes [1,5]. Hence, they are also called Type-II smectic
liquid crystals. In this analogy, the twist of a cholesteric phase penetrates into the SmA structure
through a lattice of screw dislocations, which is similar to the magnetic flux lines penetrating a Type-II
superconductor via a lattice of vortices. The detailed structure was theoretically calculated by Renn
and Lubensky [5]. A schematic representation of the arrangement of smectic-A blocks of the TGBA
phase is shown in Figure 3. The parallel screw dislocations originate due to the rotation of successive
smectic slabs relative to each other. The well-defined SmA blocks are separated by regularly-spaced
planar arrays of parallel screw dislocations. The interaction between the dislocations forms grainboundaries, and the interaction among the grain boundaries stabilizes the TGBA structure. The typicaldistance between two dislocations in a grain boundary is denoted as ld , and the inter-grain boundary
distance as lb and usually ld ∼ lb , and they are in the range of 20–30 nm [7]. Later, several variantsof the twist grain boundary phase were observed such as TGBC [8], TGBC* [9], UTGBC* [10,11], etc.
UTGBC* has a three-dimensionally-modulated structure, and it was discovered in a binary mixture ofa chiral and a nonchiral compounds.

(a)

(b)

TGBA
Figure 3. (Color online) (a) Schematic diagram showing the structure of the Twist Grain Boundary-A(TGBA ) phase. The distance between two dislocations and two grain boundaries are ld and lb
respectively. (b) A typical texture of the TGBA phase with relatively large pitch in the planar surface
anchoring condition. The different colors indicate the different twist regions formed during coolingfrom the N* phase. Reproduced with permission from [12].

1.2. Blue Phases
Blue phases are another type of frustrated phase observed in highly chiral liquid crystals.
They appear between the Isotropic (I) and cholesteric (N* ) phase transition, and the temperature
range in pure compounds is very narrow (∼1 K). In the early investigations, they appeared blue,
hence being known as the blue phase. The defect-lattice structure and POM textures of BP-I and
BP-II are shown in Figure 4. In the cholesteric phase, the orientation of the local director cantwist around two perpendicular axes, and the corresponding deformed structure is named as a“double-twist cylinder” [1]. The double twist cylinders fill up the volume by stabilizing a latticeof defect-disclinations [13,14]. The defect lines arrange themselves forming different structures,

Fluids 2018, 3, 26

4 of 15and depending on their local arrangement, they are classified into Blue Phase (BP)-I, BP-II and
BP-III. The BP-I, BP-II and BP-III have body-centered cubic, simple cubic and amorphous structure,
respectively. The symmetry of BP-III is almost similar to the isotropic phase, hence it is very difficult
to distinguish from the isotropic phase under POM. BP-I and BP-II can be identified under POM
rather easily. The stability of BP-I increases with increasing chirality, whereas BP-II prevails up to acertain range of the temperature-chirality variation. There are a few methods known to increase thetemperature range of BPs such as by dispersion of the polymer, nanoparticles and mixing of manyliquid crystals and chiral dopant. Among them, polymer-stabilized BPs are very popular, and they
were discovered by Kikuchi et al. [15]. They showed that the range can be enhanced to more than
60 K and down to room temperature. This created immense interest in the field of liquid crystals, andconsequently polymer-stabilized liquid crystal displays (LCDs) with superior display performance
were reported [16–19]. There are many studies that have reported on the complex structure and opticalproperties of blue phases. Nevertheless, several fundamental aspects of these exotic phases are yet to
be explored. For example, recently, several theoretical predictions were made on the defect dynamics
and rheological properties of BP-II and BP-I [20–24], which are yet to be explored experimentally.
2. Experimental Section
Most of the rheological measurements, unless specifically mentioned, reported here were
performed by a controlled strain Rheometer (MCR 501, Anton Paar, Graz, Austria) with cone-plateor parallel plate measuring systems. A Peltier temperature controller with an accuracy of 0.1 ◦ C wasused to vary the temperature. All the measurements were made on cooling the sample from theisotropic phase. The samples were presheared at the shear rate of 10 s−1 for 400 s before starting themeasurements. The details of the rheomicroscopy setup are given in [25,26].
3. Rheological Properties of the Twist Grain Boundary-A Phase
In pure compounds, the temperature range of the TGBA phase is usually very short (1–2 ◦ C).
Nevertheless, some attempts were made to characterize the rheological properties. Asnacios et al.
studied the rheological properties of Cholesteryl Tetradecanoate (CT) and Cholesteryl Nonanoate(CN) and focused on the rheological properties near the N* -SmA transition [27]. From thetemperature-dependent viscosity and shear rate-dependent shear stress, they found a short-range
smectic-like behavior just above the SmA-N* transition (T = TN* -SmA + 1 ◦ C). They conjectured it to
be a TGB-like phase. Later, Rajeswari et al. also studied the rheological properties of CN and its
mixture with 40 -octyloxy-4-cyanobiphenyl (8OCB ) [12]. They confirmed that indeed, a short-range
smectic-like order in the cholesteric phase exists, and it was further enhanced by mixing a small
amount of 8OCB compound. Figure 5 shows the temperature-dependent shear viscosity of the mixture
with 20 wt % 8OCB. Apart from a sharp peak in the viscosity due to the blue phases, they observed
another anomalous peak across the N* -SmA transition. The latter one was attributed to the SmA to
TGBA phase transition. However, the narrow temperature range has always been a major problem tounderstand and correlate its physical properties with the structure.
Recently, Rasmita et al. studied the rheological properties of a TGBA phase with a wide
temperature-range [25]. It was discovered in a binary mixture of two compounds, namely
4-(20 -methyl butyl phenyl)40 -n-octylbiphenyl-4-carboxylate (CE8 ) and 2-cyano-4-heptyl-phenyl-40 pentyl-4-biphenyl carboxylate (7(CN)5), where CE8 is a chiral and 7(CN)5 is a nonchiral compound,by Pramod et al. [10,11]. Sixty-three-point-six weight percent of CE8 and 36.4 wt % of 7(CN)5 in the
mixture are known to show the following phase transitions: I 123 ◦ C N* 83.5 ◦ C TGBA 60.2 ◦ C SmC * .
Thus, the temperature range of the TGBA phase is about 20 ◦ C. The phase transitions were identified
from the temperature-dependent shear viscosity measurements (Figure 6). The shear viscosity of the
TGBA phase is two orders of magnitude larger than that of the cholesteric phase.

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Figure 4. Schematic diagrams of the BP-II and BP-I. (a,c) Spatial arrangements of double twisted
cylinders in unit cells; (b) simple cubic unit cell of the disinclination lattice in BP-II; (d) body-centeredcubic unit cell of the disinclination lattice in BP-I; the black lines in (b,d) are the defect-disinclinations;(e,f) typical textures of BP-II and BP-I observed under a polarizing optical microscope (POM).
Reproduced with permission from [26].
TGBA

BPs

Figure 5. Temperature-dependent shear viscosity at different shear rates in a binary mixture of 80 wt %Cholesteryl Nonanoate (CN) and 20 wt % 40 -octyloxy-4-cyanobiphenyl (8OCB ). The black bar shows
the narrow temperature range of the induced TGBA phase. Measuring device: cone-plate with a plate
diameter of 25 mm and a cone angle of 1◦ . Reproduced with permission from [12].

Figure 7a shows the shear rate-dependent shear stress of different phases of the mixture. At alow shear rate, where γ̇ → 0, the shear stress tends to reach a constant value showing an apparent
yield stress (σy ). Figure 7b shows the variation of yield stress at different phases. In the N* phase, σyis negligibly small (10−2 Pa). In the TGBA phase, σy ' 10 Pa. Thus, the yield stress of TGBA is threeorders of magnitude larger than that of the N* phase. However, the apparent yield stress of the TGBA
phase of the studied mixture is much larger than that known for low molecular weight smectic liquid
crystals [28,29].

Fluids 2018, 3, 26

6 of 15TGB(mPa s)
eff

10

SmC *

A

3

*

2

10

N

I

1

10

60

70

80

90

100 110 120oTemperature ( C)

Figure 6. (Color online) Variation of effective shear viscosity (ηe f f ) as a function of temperature at aconstant shear rate 100 s−1 . Measuring device: cone-plate with a plate diameter of 25 mm and a coneangle of 1◦ . Reproduced with permission from [25].

Figure 7. (Color online) (a) Shear rate-dependent effective shear stress (σe f f ) at different temperatures
corresponding to different phases; (b) yield stress (σy ) at different temperatures. Measuring device:cone-plate with a plate diameter of 25 mm and a cone angle of 1◦ . Reproduced with permission
from [25].
000

The frequency-dependent shear modulus (G and G ) provides the dynamic response of the0sample. Figure 8 shows some representative frequency dependence of storage (G ) and loss moduli
00(G ) in the N* and TGBA phases. The role of defects and disorder on the viscoelastic properties ofrandomly-oriented Type-I smectic liquid crystals has been studied by many authors [4,30–36]. In thereported systems, the defects were either stabilized by quenched disorder or by dispersing colloidal
0particles in the liquid crystals. The storage modulus (G ) of such a system shows a power-law behavior:
0

G (ω ) = G0 + βω α(3)Fluids 2018, 3, 26

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where G0 is the plateau modulus arising from the zero frequency shear modulus, ω is the angularfrequency, β is a constant and α is an exponent. The TGBA is composed of rotating SmA blocks
separated by grain boundaries made of parallel screw dislocations. These defects are inbuilt into thestructure, and hence, cannot disappear due to the action of shear. The contribution of defects anddisorder to the rheological properties of the TGBA is analyzed by the same model. α obtained from the000
fitting of G (ω) and G (ω) at different temperatures is shown in Figure 9. In the N* phase, α obtained
000from the fitting of G and G is about two and one, respectively. The exponent α decreases in the TGBA
0phase and is almost equal to α ' 0.5. The solid-like behavior reflected by a plateau in G in the terminal
zone has been studied and characterized theoretically in other mesophases, such as carbonaceous
mesophases [37]. The storage modulus of liquid crystalline systems with a defect network has been
experimentally studied by Ramos et al. [38]. They showed that the contributions from the disoriented
part of the sample and regions of the sample where the layers are parallel to the shear direction are00given by G ∝ ω 1/2 and G ∝ ω 2 respectively. Similarly the loss modulus has a contribution both from00
00
the disorientated parts (G ∝ ω 1/2 ) of the sample and a Maxwell fluid type contribution (G ∝ ω).
000
In the TGBA phase, α obtained from the fitting of G and G is equal and given by α ' 0.5. Hence, thestorage and loss moduli have a major contribution from the disorientated layers of the TGBA phase.

000

Figure 8. (Color online) Frequency dependence of storage G (solid symbols) and loss G (open symbols)
moduli at two representative temperatures in the N* and TGBA phases. The solid lines are the best fits
to Equation (3). Measuring device: cone-plate with a plate diameter of 25 mm, a cone angle of 1◦ and
γ = 1. Reproduced with permission from [25].

Figure 9. (Color online) The temperature variation of exponent α obtained by fitting the power-law
equation (Equation (3)). The data corresponding to solid squares are obtained by fitting the storage
000
modulus (G ), and open squares are obtained by fitting the loss modulus (G ). Reproduced withpermission from [25].

Fluids 2018, 3, 26

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According to the soft glass rheology model, the glass transition is characterized by the power-law
0
exponent, i.e., G ∼ ω x−1 . In this model, “x” is an effective noise temperature that varies within
1 < x < 2 and is related to the exponent α by α = x − 1 [39]. The system approaches the glasstransition if x → 1, i.e., α → 0. In reference to Figure 9, across the N* to TGBA phase transition,x decreases from three to about 1.5, signifying that the rheological response of TGBA is very close to
many soft glassy materials.
The temperature variation of G0 is directly related to the defects. The variation of G0 with reduced
temperature χ ≡ (TN* -TGBA −T)/(TN* -TGBA ) is shown in Figure 10. In the TGBA phase, G0 is more than
three orders of magnitude larger than the N* phase (Figure 10). In connection to the rubber elasticity,
the elastic response of static defect network G0 varies as G0 ≈ τ/d2 , where τ is the line tension andd is a typical average spacing between defects [38]. The defect line tension τ can be expressed as
τ = Bb4 /128π 3 rc2 , where B is the layer compression modulus, b = md0 is the Burger vector of integer
strength m and rc is the defect core radius [40]. There are many smectic liquid crystals in whichB ∼ χ0.4 [41] and rc−2 ∼ ψ2 ∼ χ0.5 , where ψ is the smectic order parameter [42]. Hence, the defectline tension (τ) varies as τ ∼ χ0.9 . This implies that the elasticity of the static defect network varies as
G0 ∼ χγ with γ ≈ 0.9 [42]. Figure 10 shows the variation of G0 with the least square fit parameter γ = 1.
Assuming a medium value of Berger’s vector of integer strength m = 2 [43], B ≈ 2 × 106 Pa [30,41]
and using the relation G0 ≈ τ/d2 , the calculated inter-defect spacing in the TGBA phase is d ≈ 26 nm.
This calculated value is closely comparable with the experimentally-measured dislocation spacing in
the TGBA phase [7].

Figure 10. (Color online) Variation of G0 with reduced temperature: χ ≡ (TN* -TGBA −T)/(TN* -TGBA ).
The solid line shows the best fit to G0 ' χγ , with γ = 1. Reproduced with permission from [25].

4. Rheological Properties of Blue Phases
Recently, several theoretical predictions are made on the rheological properties of bluephases [20–22]. However, experimentally, it is much less explored due to its narrow range oftemperature. The rheological properties of BPs with a narrow range of temperature in CN werefirst reported by Kleiman et al. [44]. It exhibits the following phase transitions: I 90.52 ◦ C BP-I 91.00 ◦ C
BP-II 91.08 ◦ C BP-III 91.22 ◦ C N* . It shows three blue phases, and the temperature ranges of BP-III,
BP-II and BP-I are 14 mK, 8 mK and 48 mK, respectively. Figure 11 shows the storage modulus andviscosity as a function of temperature. They showed that BP-I and BP-II are viscoelastic solids and that
BP-III has cholesteric-like order.

Fluids 2018, 3, 26

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Figure 11. The elastic modulus and viscosity of CN at three different frequencies across the Isotropic(I)-N* transition showing three blue phases. Measuring device: torsional oscillators configured as cup
viscometers with a cup diameter of 0.51 cm. Reproduced with permission from [44].

Very recently, Rasmita et al. made detailed experimental studies on the rheology of BPs by taking
a mixture of four fluorinated compounds and a highly chiral dopant [26,45]. It exhibits the following
phase transitions: I 46.4 ◦ C, BP-II 41.4 ◦ C BP-I 36.6 ◦ C N* . The temperature ranges of BP-I and BP-II are
4.8 ◦ C and 5 ◦ C, respectively, and much larger than the CN compound. The temperature-dependentshear stress at a fixed shear rate of the mixture is shown in Figure 12. Distinct changes of stress (τ)
across the phase transitions are clearly observed. A small kink in the stress-curve marks the BP-II
to BP-I phase transition. The shear stress of BP-I is larger than that of BP-II and N* phase. This isconsistent with the previous report, i.e., the elasticity of the BP-I is greater than that of BP-II.

Figure 12. (Color online) Temperature-dependent shear stress at a constant shear rate (10 s−1 ). Threedifferent phase transitions are marked by dotted vertical lines. Measuring device: cone-plate with aplate diameter of 25 mm and a cone angle of 1◦ . Reproduced with permission from [26].

Simultaneous optical rheomicroscopy measurements reveal the change in the textures across thephase transitions. For example, Grandjean–Cano lines are observed in the BP-II phase, and they arealigned perpendicular to the shear direction (Figure 13a). The increase of shear stress as compared tothe isotropic phase is due to the motion of the Grandjean–Cano lines. With decreasing temperature,some filamentary structures appear in the BP-I phase. This corresponds to the increase in shear
stress in the BP-I phase as compared to the BP-II phase (Figure 13b). These filamentary structures
are continuously broken and reconnected under steady shear. The N* phase (Figure 13c,d) showsFluids 2018, 3, 26

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a typical texture of oily-streak defects. Dupuis et al. have studied the rheology of cholesteric bluephases by computer simulation [22]. It was predicted that the shear forces can bend, twist and unlock
the disclination lines. Under low shear, the defect network opposes the flow; hence, the apparent
shear viscosity and shear stress become large. However, under high shear, they show shear thinningbehavior signifying the disruption of the defect network.

45.5oC(a)38.5oC
(b)

BP-II

BP-I

17.3oCo21.5 C
(c)

(d)

N*

N*

Figure 13. Optical rheomicroscopy images taken during the measurement of temperature-dependentshear stress at a constant shear rate of 10 s−1 in three different phases. The direction of shear is shownby an arrow. P and A indicate Polarizer and Analyzer. Measuring device: rheomicroscopy setup
with a parallel plate of diameter 43 mm. The gap between the two parallel glass plates is 0.075 mm.
Reproduced with permission from [26].

The flow behaviors of the BP-II, BP-I and N* phases are shown in Figure 14. The N* phase exhibitsa typical shear thinning behavior. The yield stresses of BP-II and BP-I are about one order of magnitudelarger as compared to the N* phase. BP-I and BP-II both exhibits multiple shear thinning behavior
at a higher shear rate range. The shear stress of BP-II shows three distinct regimes, i.e., BP-II(1):γ̇ = 0.01–0.3 s−1 ; BP-II(2): γ̇ = 0.3–2 s−1 and BP-II(3): γ̇ = 2–1000 s−1 . Figure 14b shows a power-law fit;
τ = aγ˙α , which describes the data to a very good approximation. The fit parameters are: a = 0.87 and
α = 0.33 in BP-II(1); a = 0.42 and α = 0.35 in BP-II(2); a = 0.12 and α = 0.87 in BP-II(3).

Figure 14. (Color online) (a) Shear rate-dependent shear stress at three different temperatures
representing the BP-II, BP-I and N* phases; (b) red lines are best fits to the power-law: τ = aγ˙αin the BP-I phase. Measuring device: cone-plate with a plate diameter of 25 mm and a cone angle 1◦ .
Reproduced with permission from [26].

Fluids 2018, 3, 26

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Rheomicroscopy images at some representative shear rates are shown in Figure 15. In a low shear
rate regime (0.01–0.3 s−1 ), in the BP-II phase, the Grandjean–Cano lines flow parallel to the shear
direction (downward green arrows in BP-II). In the intermediate shear rate region (0.3–2 s−1 ), theseGrandjean–Cano lines are broken, and at a higher shear rate-range (2–1000 s−1 ), the system shows aflow-aligned nematic-like state. The BP-I also shows multiple shear thinning regimes, but they are notclearly separable like BP-II. However, it is clear that multiple shear thinning is due to the breaking of
texture at different shear rates as shown in Figure 15 (see BP-I). The N* phase shows uniform shear
thinning behavior, and there are no notable changes observed in the textures up to the shear rate of
10 s−1 . Above this shear rate, probably the helix uncoils, and the sample flows like nematic, causing adecrease in the viscosity. Both BP-II and BP-I also exhibit the flow-induced nematic texture at a shear
rate much higher than 200 s−1 .

BP-II
0.01s-1

0.1s-1

3.16s-1

0.01s-1

0.316s-1

3.16s-1

0.01s-1

0.316s-1

3.16s-140s-1

794s-140s-1794s-140s-1794s-1BP-I

N*

Figure 15. Rheomicroscopy images taken between two crossed polarizers during the measurementof the shear rate-dependent shear stress in three different phases. The direction of shear is shownby an arrow on the right-hand side. The downward green arrows in BP-II show the motion of theGrandjean–Cano lines. Measuring device: rheomicroscopy setup with the parallel plate of a diameter of
43 mm. The gap between two parallel glass plates is 0.075 mm. Reproduced with permission from [26].

Recently, Henrich et al.
studied the rheology of cubic blue phases by computer
simulation [20,21,23]. They have identified various flow regimes with increasing shear rate in both
BP-I and BP-II. The typical flow curve in both phases is shown in Figure 16. Experimental resultsof Figure 14 qualitatively agree with the predictions of the simulation except that shear rate ranges
are different.

Fluids 2018, 3, 26

12 of 15(a) BP-II(b) BP-I

Figure 16. Flow curve of shear rate-dependent shear stress showing various flow regimes of(a) BP-II and (b) BP-I. Above a critical shear rate, the network breaks up into the Grandjean texture(open triangles) or a flow-aligned nematic state at even higher shear rates (solid triangles). The redarrows indicate a fall in the shear stress values. In the simulation, a simple shear flow was imposed bymeans of the Lees–Edwards boundary conditions with the top (bottom) part of the system flowing in
the positive (negative) x-direction and the velocity gradient along the y-direction. The shear rates werevaried over more than three orders of magnitude. Reproduced with permission from the [21].

The deformation behavior of blue phases is further studied by the amplitude sweep. The results
000of amplitude sweep, i.e., the strain-dependent storage (G ) and the loss (G ) moduli at different
temperatures representing different phases (BP-II, BP-I and N* ), are shown in Figure 17. The N* phaseshows a typical fluid-like behavior. The shear moduli of both the blue phases (BP-II and BP-I) are0
higher than that of the N* phase. The storage modulus (G ) of BP-I is about five-times larger than thatof BP-II. The Linear Visco-Elastic (LVE) ranges of both BP-II and BP-I are almost equal, i.e., γLVE = 4%.
000
The strain-induced fluidization (the crossover of G and G ), or the critical strain amplitude denotedby γc , is 16% in the case of BP-I. However, in BP-II, it occurs at a slightly higher strain value (γc = 20%)than that of BP-I. It was interpreted that in BP-II, the defect lines intersect with each other, which needsa larger strain amplitude to break.

G', G" (Pa)

10

1

G'

o

BP-II (43 C)

G"

0.1

G'
G"
G'

o

BP-I (39.5 C)o

N* (15 C)

G"

0.01
0.01

0.1

1

(

10

100

)
0

00

Figure 17. (Color online) The strain dependence of the storage (G ) and the loss (G ) moduli of thesample at three different temperatures at ω = 1 rad s−1 . Measuring device: cone-plate with a plate
diameter of 25 mm and a cone angle of 1◦ . Reproduced with permission from [26].

The dynamics of the defect networks are investigated by a step-strain measurement. Figure 18ashows the shear stress response with time under the applied step strain. The applied strain is changed
with time stepwise from 0.3–30% and then again back to 0.3%. Therefore, there are three regionswith two steps in which 0.3% is within the LVE range and 30% is above the LVE range. The durationof each step is 600 s. The N* phase shows a typical stress response under applied step strain in all
the regions, as expected. Both BP-II and BP-I exhibit different responses in Region-II and Region-III
than the cholesteric phase. In Region-II of BP-II, a gradual decrease of shear stress with time isFluids 2018, 3, 26

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observed. It could be due to the breaking of defect nodes under high strain followed by a continuous
flow-aligned state. In Region-II of BP-I, a constant shear stress response is observed. BP-I showssinusoidal oscillations of shear stress with time in Region-III that decays with time. Figure 18b shows
the periodic modulation of stress with time in Region-III of BP-I. Stress oscillation after cessation ofshear in chiral liquid crystals has been simulated and characterized for cholesteric liquid crystals[46]. The oscillation in shear stress of BP-I is attributed to the lattice distortion, i.e., displacement ofdefects from the equilibrium positions. The equilibrium structure tends to be restored upon removalof the strain, which gives rise to periodic stress oscillation, which decays with time [25]. Hence, thestep strain experiment reveals the different dynamical behavior of BP-I and BP-II due to their distinct
defect structures.

Figure 18. (Color online) (a) Time-dependent shear stress at three different temperatures forω = 1 rad/sec; (b) enlarged area of BP-I in Region-III (see (a)) showing the periodic stress oscillation.
Measuring device: cone-plate with a plate diameter of 25 mm and a cone angle of 1◦ . Reproduced withpermission from [26].

5. Conclusions
Rheological properties of two liquid crystalline intermediate phases in highly chiral liquid crystals
namely TGBA and BPs are presented. Topological defects are inherent to both phases and contributeto the rheological properties. The analysis of the complex shear modulus shows that the elasticity of
the TGBA phase is due to the contribution of inbuilt structural defects. The TGBA liquid crystals area defect-mediated soft solid whose dynamics is similar to soft glassy materials. Simple calculations
based on the theory of rubber elasticity provide the inter-dislocation distance. Both BP-II and BP-I show
several flow regimes with characteristic microstructures. The experimental results are qualitatively
in agreement with some recent results of computer simulation. Both blue phases exhibit a solid-likebehavior, while the cholesteric phase shows a gel-like behavior. The sinusoidal stress oscillations in
BP-I after the removal of strain are connected to the dynamic response of the distinct defect structures.
The critical strain amplitude for strain-induced fluidization, i.e., γc of BP-II is larger than that of BP-I,
and it is attributed to the defect structure; in particular, the breaking of defect nodes in BP-II requires
larger strain than BP-I. Finally, it may be mentioned that there are several predictions on the dynamic
behavior of defects in the blue phases from the computer simulations, and to validate them, moreexperiments are required.

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Acknowledgments: We gratefully acknowledge the support from Depart of Science and Technology (Project Ref.:
DST/SJF/PSA-02/2014-2015).
Conflicts of Interest: The authors declare no conflict of interest.

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