PHYSICAL REVIEW MATERIALS 4, 115601 (2020)

Smecticlike rheology and pseudolayer compression elastic constant of a twist-bend
nematic liquid crystal
M. Praveen Kumar ,1 P. Kula,2 and Surajit Dhara

1,*1School of Physics, University of Hyderabad, Hyderabad 500046, India
2
Institute of Chemistry, Faculty of Advanced Technologies and Chemistry, Military University of Technology, Warsaw, Poland(Received 18 August 2020; revised 7 October 2020; accepted 13 October 2020; published 16 November 2020)
In twist-bend nematic (NTB ) liquid crystals (LCs), the director (mean molecular orientation) exhibits heliconical structure with nanoscale periodicity. On the mesoscopic scale, NTB resembles layered systems (like smectics)
without a true mass density wave, where the helical pitch is equivalent to a “pseudolayer.” We study rheologicalproperties of a NTB phase and compare the results with those of a usual smectic-A phase. Analyzing the shear
response and adapting a simplified physical model for the rheology of defect-mediated lamellar systems, we measure the pseudolayer compression elastic constant Beff of the NTB phase from the measurements of the dynamic
modulus G∗ (ω). It is found that Beff of the NTB phase is in the range of 103 –106 Pa and it follows a temperaturedependence, Beff ∼ (TTB − T )2 , as predicted by the recent coarse-grained elastic theory. Our results show thatthe structural rheology of NTB LCs is strikingly similar to that of the usual smectic LCs, although the temperaturedependence of Beff is much faster than that of smectic LCs as predicted by the coarse-grained models.
DOI: 10.1103/PhysRevMaterials.4.115601

I. INTRODUCTION

Experimental discovery of the twist-bend nematic (NTB )phase in bent-core liquid crystals has created immense interestin the liquid crystal community [1–7], although it was theoretically predicted much earlier from different perspectives[8–11].
In the NTB phase, the director n̂ (the mean molecular orientation) exhibits periodic twist and bend deformations forming
a conical helix and is tilted with respect to the axis L, asshown in Fig. 1(a). The typical pitch p of the heliconicalstructure is of the order of 10 nm, thus comparable to a few
molecular length. Commonly, the NTB phase is observed in
odd-membered liquid crystal dimers wherein two mesogenic
units are connected through a flexible spacer [12–14].
A fascinating feature of the NTB phase is the observationof spontaneous chirality i.e., formation of both left- and righthanded helical domains even though the constituent molecules
are achiral. This leads to several unusual physical propertiesof the NTB phase compared to the conventional nematic phase(N) [15–21].
A few coarse-grained theories have been proposed to
explain the emergence of the NTB phase from the hightemperature uniform nematic phase. Meyer and Dosov
showed that the elastic properties of the NTB phase could
be viewed in two different length scales in reference to thepitch length p [22]. When the considered length l is lessthan p, i.e., l < p, the elastic description is similar to thatof the usual nematics. On the other hand, when l  p, theelastic description is similar to regular lamellar systems suchas cholesteric and smectic liquid crystals (LCs) [22]. In thelatter picture, the thickness of one pitch can be considered a*

sdsp@uohyd.ernet.in

2475-9953/2020/4(11)/115601(6)pseudolayer, and the large-scale elasticity of the NTB phasecan be described in terms of an effective pseudolayer compression elastic constant Beff and an curvature elastic constantN, and the corresponding free-energy density of NTB can be
K11
expressed as [22]


1
1 N 1
1 2
2
fTB = Beff  + K11
+
,
(1)
2
2
R1
R2
where  is the pseudolayer compression and R1 and R2 are
the curvatures. Another coarse-grained theory was developed
considering the helipolar order and it’s coupling with bend
distortions [20,23]. Both theories predicted that the temperature dependence of Beff is much faster than that of the usualsmectic-A (Sm-A) LCs.
There have been very few experimental studies on the measurements of Beff of NTB LCs [23–25]. For example, Gorecka
et al. measured Beff of CB7CB and some chiral NTB LCs.
Their reported values are in the range of a usual Sm-A LC(106 –107 Pa) and vary inversely with the temperature [24].
Parsouzi et al. reported that Beff is in the range of 103 –104 Pa
and it scales as Beff ∼ (TTB − N )3/2 [23]. Thus, a several orders of magnitude difference in the reported values, measured
on two different samples using two different methods andits universal temperature dependence, is still an unresolved
problem. In this paper we report experimental studies on therheological properties of a NTB LC. We use a method formeasuring Beff from the dynamic shear modulus G∗ (ω). We
discuss the temperature dependence of Beff and compare itwith that proposed by the coarse-grained elastic theories of
the NTB phase.
II. EXPERIMENTAL

The LC material 1,ω-bis(4-cyanobiphenyl-4 -yl) alkane(CB9CB) was synthesized in our laboratory. It is a115601-1©2020 American Physical Society

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PHYSICAL REVIEW MATERIALS 4, 115601 (2020)

FIG. 1. (a) Schematic view of the heliconical molecular orientation of the nematic twist-bend (NTB ) phase. p represent helicalpitch equivalent to pseudolayer thickness. (b) Chemical structure of
CB9CB molecules used in the study. (c) Polarizing optical microscope texture at 94 ◦ C in the NTB phase. L is the macroscopic averageorientation of n̂ over several periods.

cyanobiphenyl-based dimer with an odd number of methylene
units (n = 9) in the flexible spacer [Fig. 1(b)]. It exhibits thefollowing phase transitions: isotropic I 124 ◦ C to N 108 ◦ C
to NTB 84 ◦ C to crystalline (Cry) and an enantiotropic transition with the widest temperature range of the NTB phase
22 ◦ C among the LCs in the homologous series of CBnCB[14]. For the purposes of comparison we also studied 8CB(octylcyano biphenyl), which shows the following phase transitions: I 41 ◦ C to N 34 ◦ C to Sm-A 22 ◦ C to Cry. We useda strain-controlled rheometer (MCR 501, Anton Paar) with acone-plate measuring system with a plate diameter of 25 mm
and cone angle of 1◦ for rheological measurements. A Peltiertemperature controller was attached to the bottom plate tocontrol the temperature with an accuracy of 0.1 ◦ C. A hood
was used to cover the measuring plates for uniformity of thesample temperature. Temperature-dependent viscosity wasmeasured in cooling the sample from the isotropic phase. To
measure the dynamic shear modulus the sample was quenchedfrom the isotropic to the NTB phase at a rate of 15 ◦ C/min. A
total of 5 g of LC was synthesized, and about 200 mg wereused for each rheological measurements. Initially, the phasetransitions and textures were observed using a polarizing optical microscope (Olympus BX51) and a temperature controller(Mettler FP 90). A typical texture of an unaligned sample isshown in Fig. 1(c). It is noticed that the texture of NTB is very
similar to that of the focal conic textures of the usual Sm-A
LCs.
III. RESULTS AND DISCUSSION

To begin with we measure the shear viscosity of CB9CB
as a function of temperature at two shear rates (γ̇ = 100 and
10 s−1 ) to identify the phase transition temperatures. Since the

FIG. 2. (a) Temperature-dependent effective viscosity ηeff of
CB9CB (blue squares) and 8CB (black spheres) LCs at two different
shear rates, namely, γ̇ = 100 s−1 (squares) and γ̇ = 10 s−1 (circles).
Inset: Variation of ∂η∂Teff with temperature T of CB9CB. (b) Shearrate-dependent effective shear stress σeff at fixed temperatures. The
solid lines are theoretical fits to Eq. (2).

orientation of the director with respect to the shear direction
usually changes with temperature, we define it as effective
viscosity ηeff . As shown in Fig. 2(a), the N to NTB transition
is identified from the rapid increase (more than two ordersof magnitude) of ηeff with respect to the N phase. The onsetof the N-NTB transition (108 ◦ C) is better seen in the inset of
Fig. 2(a). We also measured the temperature-dependent ηeffof 8CB LC, as shown in Fig. 2(a). It is evident that the magnitude and the overall temperature dependence of ηeff of thetwo samples are very similar. This similarity also presumably
indicates that the pretransitional fluctuations and the resulting
director dynamics across the N-NTB transition are similar tothat of the N-Sm-A transition [26–28]. In analogy with 8CB
LC, the three simplest orientations of the pseudolayers canbe considered, wherein the layer normals are parallel to thevorticity ∇ × v, velocity gradient ∇v and flow directions v, asshown schematically in Fig. 3. These are commonly known as
perpendicular, parallel, and transverse orientations. The largeηeff of the NTB phase is expected to arise from the transverse
orientation of the pseudolayers, similar to those reported in
the Sm-A phase of 8CB LC [29].

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PHYSICAL REVIEW MATERIALS 4, 115601 (2020)

FIG. 3. Schematic representation of pseudolayer orientations ina shear flow.

The flow curve of NTB phase has been studied and compared with that of the Sm-A phase of 8CB, as shown in
Fig. 2(b). The NTB phase shows a yield stress similar to thatof the Sm-A phase of 8CB LC. However, the rheology of the
Sm-A phase of 8CB is complex as it shows a shear-induced
structural transition [30]. A small but discontinuous change
in σeff near γ̇ = 8 s−1 in Fig. 2(b) and an increase in ηeff
at 24◦ C [Fig. 2(a)] could be a signature of such an effect.
Interestingly, a similar discontinuity is observed in the case of
CB9CB at γ̇ = 22 s−1 . Further studies are required to confirm
the occurrence of such a transition in CB9CB. Nevertheless,the shear-rate-dependent shear stress σeff is fitted with the
Herschel-Bulkley (HB) model:
σeff = σy + Aγ̇ n ,(2)

where σy is the yield stress and A and n are constants. The fit
parameters obtained are n = 0.71, σy = 1.0 Pa, A = 1.9 for
NTB and n = 0.66, σy = 7.1 Pa, A = 2.9 for the Sm-A phase.
The two sets of fit parameters characterizing the flow curvesof the two phases are reasonably close, suggesting they have
similar flow behaviors. These two samples have structuralsimilarity; namely, the pseudolayer thickness (∼10 nm) of
the twist-bend nematic is close to the layer thickness of 8CB(∼2 nm). Moreover, both samples exhibit focal conic textures.
Hence, their generic mechanical responses under shear are
similar.
As a next step, we measure the dynamic modulus G∗ (ω) =G (ω) + iG (ω). The regime of linear viscoelasticity of NTB isdetermined by performing oscillatory measurements in which
the strain amplitude varies from γ = 0.01% to γ = 100% at
a fixed frequency ω = 1 rad/s. The strain amplitude dependence can be described by the empirical relation [31]G (ω, γ ) =G (ω, 0),
1 + γ /γc(3)where γc is the critical strain amplitude. As shown in Fig. 4(a),the critical strain amplitude γc = 3.3%, and the modulusG (1, 0) = 391 Pa, setting the upper limit of the linear viscoelastic regime. For comparison we also measured G∗ of the
Sm-A phase of 8CB, shown in Fig. 4(a). It is interesting to notethat not only the strain dependence but also the magnitudesof the shear moduli of NTB and Sm-A are comparable, indicating they have common structural origin. Further, we havemeasured the temperature dependence of dynamic moduli of
NTB at a fixed shear amplitude γ = 0.1% and observed thatG > G in the entire NTB phase [Fig. 4(b)], which is remarkably similar to that is observed in smectics with a true mass

FIG. 4. (a) Strain amplitude dependence of the storage G (solid
symbols) and loss moduli G (open symbols) for NTB (T = 87 ◦ C)
and 8CB (T = 26 ◦ C). The solid line is a theoretical fit to Eq. (3).
Arrows indicate crossover strains. (b) Temperature dependence of G
(solid symbols) and loss moduli G (open symbols) at a fixed strain
amplitude γ = 0.1%. Measurements are performed at frequency
ω = 1 rad/s.

density wave. Hence, the shear response of the NTB phase canbe discussed in analogy to the rheological responses of theusual Sm-A liquid crystals [31]. Like Sm-A, NTB is solidlike
in one direction and liquidlike in the other two directions. The
three simplest orientations of the pseudolayers are considered,as shown in Fig. 3. In perpendicular and parallel orientations
the pseudolayers can slide past each other easily, and the NTB
phase behaves like a liquid. In the transverse orientation theshear tends to change the pseudolayer spacing. As a result,the NTB phase shows a viscoelastic solidlike behavior, andconsequently, G > G .
The observed solidlike response of the NTB phase can be
explained based on a simple physical model described for
defect-mediated cholesteric and smectic LCs. According tothis model, the storage modulus can be expressed as [32]G (ω) = G0 + βd ω1/2 + β0 ω2 .

(4)

The first term, G0 , arises from the elasticity of the static
defects [33]. The second term, βd ω1/2 , arises from the regions of misaligned pseudolayers in the sample [34]. The
last term results from the regions of the sample where thepseudolayers are parallel to the shear direction. The propor2
tionality constants√ √ are given by β0  η(γ1 /K )(p/4π ) and
βd = (π /24 2) (Beff η), where γ1 is the rotational viscosity,

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FIG. 5. Frequency dependence of storage G (solid symbols) andloss G (open symbols) moduli at a few representative temperatures.
Solid lines are theoretical fits of Eqs. (4) and (5) to G and G ,respectively.

η is the effective viscosity, and Beff is the compression elastic
modulus [34]. In particular, βd describes the response of thelamellar regions with the layer normal oriented such that strain
involves layer compression. Following similar arguments, theloss modulus can be written as [32]G (ω) = αd ω1/2 + α0 ω,(5)where the first and second terms arise from the misaligned
parts of the samples and the Maxwell-fluid-type contribution,
respectively. Thus, by measuring βd and α0 from the disoriented sample, we can estimate Beff . In order to measure theseparameters at different temperatures we quenched the sample
directly from the isotropic to the NTB phase to obtain a mostly
disoriented sample. Figure 5 shows some representative plotsof G (ω) and G (ω) at different temperatures in the NTB phase.
The parameters obtained by fitting Eqs. (4) and (5) at different
temperatures are shown in Fig. 6. The fit parameter β0 is
found to be very small (10−3 ) and does not vary considerably with temperature. It is noted that βd  αd , as expected
theoretically, and both increase with decreasing temperature.
Assuming the whole sample is in the disorientated state, thetemperature dependence of Beff can be expressed as √ 2βd2 24 2
Beff =.
(6)α0πFigure 7 shows the temperature dependence of calculated Beffin the NTB phase. Just below the N-NTB transition Beff isrelatively smaller, 1.5 × 103 Pa (T = 105 ◦ C), and it increases
rapidly to 2 × 106 Pa (T = 85 ◦ C). The latter value is only

FIG. 6. (a) Temperature dependence of fit parameters βd (redspheres) and αd (navy blue squares). Inset: Temperature dependenceof α0 . (b) Temperature dependence of the fit parameter G0 . Inset: The
solid line shows the fit result G0 ∼ χ , where the reduced temperatureχ = (TTB − T )/TTB .

one order of magnitude lower than the typical layer compression modulus of the Sm-A phase of 8CB [35]. So far,experimentally, Beff of very few NTB LCs have been measured. Gorecka et al. measured the temperature dependenceof Beff of a few chiral twist-bend nematic LCs, including
CB7CB, using the atomic force microscopy technique and

FIG. 7. Temperature dependence of effective elastic compressional modulus Beff . The solid line is a theoretical fit to the equation
Beff ∼ (TTB − T )α , where α = 2.0 ± 0.1. Inset: Log-log scale.

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reported that Beff is in the range of 106 –107 Pa, comparable
to an ordinary Sm-A LC. Using a dynamic light-scattering
technique, Parsouzi et al. reported that Beff of the NTB phaseof a LC made of a multicomponent mixture is in the rangeof 103 –104 Pa, which is almost three orders of magnitude
smaller than the ordinary Sm-A LCs [23,25]. Our experimentshows a wide variation of Beff , covering both ranges. Such awide variation of Beff could partly be attributed to the increasein the cone angle θ with decreasing temperature. Based on acrude model, Beff = K2 (2π /p)2 sin4 θ [25], where θ increases
with decreasing temperature [19,36]. Considering physicalparameters for CB7CB such as p = 10 nm [19] and θ = 10◦[36] (2 ◦ C below the N-NTB transition), p = 8 nm [19] andθ = 33◦ [36] (25 ◦ C below the transition), and K2 = 3 pN,
the calculated Beff near two limiting temperatures are given
by 1.1 × 103 and 1.6 × 105 Pa. The calculated Beff close to
the transition agrees reasonably well with our experiments,
but it is smaller by one order of magnitude far below thetransition. This assessment suggests that merely increasing thecone angle with decreasing temperature can enhance Beff byalmost two orders of magnitude.
The temperature dependence of Beff was predicted theoretically using coarse-grained theoretical models. In analogy
with Sm-A∗ , Meyer and Dosov defined smecticlike effectiveN
pseudolayer compression and bending elastic constants K33
N2[22]. Assuming K33 < 0, they predicted Beff ∼ (TTB − T ) .
Parsouzi et al. developed another model accounting for thehelical polarization field and its coupling with the bend distortion of the director [23]. Considering a small variation in
the pseudolayer spacing and resulting changes in the coneangle and polar order, the theory predicts that there are
two regimes of Beff . For temperature T sufficiently close to
TTB , Beff ∼ (TTB − T )3 , whereas for T sufficiently below TTB ,the theory gives Beff ∼ (TTB − T )3/2 . Experimentally, theyfound (in a mixture exhibiting the NTB phase) that withina relatively small temperature range (∼6◦ C), Beff scales as(TTB − T )3/2 .
To get an estimation of the scaling exponent of Beff , we fit
our data to the equation Beff ∼ (TTB − T )α , with α being a fit
parameter, as shown in Fig. 7. The inset shows the variationin the log-log scale. We obtain α = 2.0 ± 0.1, which is equal
to the scaling exponent predicted by the “negative elasticity”
model of Meyer and Dosov [22].
The temperature dependence of G0 as shown in Fig. 6(b)indicates that the static defects contribute to a mechanicalresponse of the NTB phase similar to that of the Sm-A phase.
The inset in Fig. 6(b) shows that G0 scales with temperature
as G0 ∼ χ , where χ = (TTB − T )/TTB is the reduced temperature. This is slightly faster than that in the case of Sm-A (χ 0.7 )[37], as expected in view of the fact that B scales much fasterwith temperature in the NTB phase.
Three remarks are in order. First, in estimating βd it has
been assumed that the pseudolayers are disoriented in thewhole sample. This assumption is reasonable as the sample
was quenched from the isotropic phase to the twist-bendphase. Nevertheless, if the sample is partly oriented we do
not expect an order of magnitude reduction of βd and, consequently, Beff . Following the same procedure, Beff of 8CB ismeasured at a few temperatures, which agrees well with what
was reported [35]. Second, the experimental results indicatethat there are defects in our system whose contribution to
the elasticity increases linearly with decreasing temperature.
This might indicate a type-II NTB having a twist grain boundary like structure as predicted by the coarse-grained model[22]. However, at this point we can say that more experimental investigations are required to draw any unambiguous
conclusion. Last, it may be mentioned that although NTB is
structurally closer to cholesterics, the mechanical responses of
NTB and smectics are very much alike. This can be attributed
to the fact that the thickness of pseudolayers in NTB is muchcloser to the smectic layer than the pitch of the usual cholesterics.

[1] V. P. Panov, M. Nagaraj, J. K. Vij, Yu. P. Panarin, A. Kohlmeier,M. G. Tamba, R. A. Lewis, and G. H. Mehl, Phys. Rev. Lett.
105, 167801 (2010).
[2] M. Čopič, Proc. Natl. Acad. Sci. U.S.A. 110, 15855 (2013).
[3] D. Chen, J. H. Porada, J. B. Hooper, A. Klittnick, Y. Shena,
M. R. Tuchbanda, E. Korblova, D. Bedrov, D. M. Walba, M. A.

Glaser, J. E. Maclennana, and N. A. Clark, Proc. Natl. Acad.
Sci. U.S.A. 110, 15931 (2013).
[4] V. Borshch, Y.-K. Kim, J. Xiang, M. Gao, A. Jákli, V. P. Panov,J. K. Vij, C. T. Imrie, M. G. Tamba, G. H. Mehl, and O. D.
Lavrentovich, Nat. Commun. 4, 2635 (2013).
[5] L. Beguin, J. W. Emsley, M. Lelli, A. Lesage, G. R. Luckhurst,

IV. CONCLUSION

In this work we have presented rheological properties ofa NTB LC. The structural rheology of the NTB phase is foundto be remarkably similar to that of the usual Sm-A phase ofcalamitic liquid crystals. Our measurements revealed that inspite of the absence of a true mass density wave, NTB LCs are
viscoelastic solids similar to many defect-mediated lamellar
systems. We found that Beff is relatively softer near the N-NTB
transition but increases with decreasing temperature to threeorders of magnitude more at a much faster rate than theusual Sm-A LCs. The temperature dependence of Beff agrees
well with the prediction of the coarse-grained elastic theories.
Thus, our results provide a valuable test of the validity ofthe proposed theoretical models. This experiment also offer
perspectives on NTB LCs and open unexplored aspects of therheology of nematic LCs with nanoscale modulation of thedirector.

ACKNOWLEDGMENTS

S.D. acknowledges financial support from SERB (Ref. No.
CRG/2019/000425) and DST-FIST-II, School of Physics.
P.K. acknowledges financial support from MUT UGB 22-760.
P.K. acknowledges UGC-CSIR for a fellowship.

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B. A. Timimi, and H. Zimmermann, J. Chem. Phys. B 116, 7940
(2012).
[6] G. Pajak, L. Longa, and A. Chrzanowska, Proc. Natl. Acad. Sci.
U.S.A. 115, E10303 (2018).
[7] J. Zhou, W. Tang, Y. Arakawa, H. Tsuji, and S. Aya, Phys.
Chem. Chem. Phys. 22, 9593 (2020).
[8] R. B. Meyer, in Molecular Fluids, edited by R. Balian and G.
Weill, Les Houches Summer School in Theoretical Physics Vol.
25 (Gordon and Breach, New York, 1976), pp. 273–373.
[9] V. L. Lorman and B. Mettout, Phys. Rev. Lett. 82, 940 (1999).
[10] V. L. Lorman and B. Mettout, Phys. Rev. E 69, 061710 (2004).
[11] I. Dosov, Europhys. Lett. 56, 247 (2001).
[12] R. J. Mandle, E. J. Davis, C. T. Archbold, S. J. Cowling, andJ. W. Goodby, J. Mater Chem. C 2, 556 (2014).
[13] M. Cestari, S. Diez-Berart, D. A. Dunmur, A. Ferrarini, M. R.
de la Fuente, D. J. B. Jackson, D. O. Lopez, G. R. Luckhurst,
M. A. Perez-Jubindo, R. M. Richardson, J. Salud, B. A. Timimi,
and H. Zimmermann, Phys. Rev. E 84, 031704 (2011).
[14] D. A. Paterson, J. P. Abberley, W. T. A. Harrison, J. M. D.
Storey, and C. T. Imrie, Liq. Cryst. 44, 127 (2017).
[15] C. Meyer, G. R. Luckhurst, and I. Dozov, Phys. Rev. Lett. 111,067801 (2013).
[16] N. Sebastian, B. Robles-Hernández, S. Diez-Berart, J. Salud,
G. R. Luckhurst, D. A. Dunmur, D. O. López, and M. R. de laFuente, Liq. Cryst. 44, 177 (2017).
[17] N. Trbojevic, D. J. Read, and M. Nagaraj, Phys. Rev. E 96,
052703 (2017).
[18] S. A. Pardaev, S. M. Shamid, M. G. Tamba, C. Welch, G. H.
Mehl, J. T. Gleeson, D. W. Allender, J. V. Selinger, B. Ellman,
A. Jakli, and S. Sprunt, Soft Matter 12, 4472 (2016).
[19] C. Zhu, M. R. Tuchband, A. Young, M. Shuai, A. Scarbrough,D. M. Walba, J. E. Maclennan, C. Wang, A. Hexemer, and N. A.
Clark, Phys. Rev. Lett. 116, 147803 (2016).
[20] Z. Parsouzi, S. M. Shamid, V. Borshch, P. K. Challa, A. R.
Baldwin, M. G. Tamba, C. Welch, G. H. Mehl, J. T. Gleeson,
A. Jakli, O. D. Lavrentovich, D. W. Allender, J. V. Selinger,
and S. Sprunt, Phys. Rev. X 6, 021041 (2016).

[21] B. Robles-Hernández, N. Sebastián, M. R. de la Fuente, D. O.
López, S. Diez-Berart, J. Salud, M. B. Ros, D. A. Dunmur,
G. R. Luckhurst, and B. A. Timimi, Phys. Rev. E 92, 062505(2015).
[22] C. Meyer and I. Dosov, Soft Matter 12, 574 (2016).
[23] Z. Parsouzi, S. A. Pardaev, C. Welch, Z. Ahmed, G. H. Mehl,
A. R. Baldwin, J. T. Gleeson, O. D. Lavrentovich, D. W.
Allender, J. V. Selinger, A. Jakli, and S. Sprunt, Phys. Chem.
Chem. Phys. 18, 31645 (2016).
[24] E. Gorecka, N. Vaupotic, A. Zep, D. Pociecha, J. Yoshioka, J.
Yamamoto, and H. Takezoe, Angew. Chem., Int. Ed. 54, 10155(2015).
[25] S. M. Salili, C. Kim, S. Sprunt, J. T. Gleeson, O. Parri, and A.
Jakli, RSC Adv. 4, 57419 (2014).
[26] C. R. Safinya, E. B. Sirota, and R. J. Plano, Phys. Rev. Lett. 66,
1986 (1991).
[27] J. Ananthaiah, M. Rajeswari, V. S. S. Sastry, and S. Dhara, Eur.
Phys. J. E 34, 74 (2018).
[28] M. Praveen Kumar, D. Venkata Sai, and S. Dhara, Phys. Rev. E
98, 062701 (2018).
[29] R. G. Larson, The Structure and Rheology of Complex Fluids(Oxford University Press, New York, 1999), p. 480.
[30] P. Panizza, P. Archambault, and D. Roux, J. Phys. II 5, 303(1995).
[31] R. H. Colby, C. K. Ober, J. R. Gillmor, R. W. Connelly, T.
Doung, G. Galli, and M. Laus, Rheol. Acta 36, 498 (1997).
[32] L. Ramos, M. Zapotocky, T. C. Lubensky, and D. A. Weitz,
Phys. Rev. E 66, 031711 (2002).
[33] R. Bandyopadhyay, D. Liang, R. H. Colby, J. L. Harden, and
R. L. Leheny, Phys. Rev. Lett. 94, 107801 (2005).
[34] K. Kawasaki and A. Onuki, Phys. Rev. A 42, 3664(R) (1990).
[35] M. Benzekri, J. P. Marcerou, H. T. Nguyen, and J. C. Rouillon,
Phys. Rev. B 41, 9032 (1990).
[36] C. Meyer, G. R. Luckhurst, and I. Dosov, J. Mater. Chem. C 3,
318 (2015).
[37] S. Fujii, S. Komura, Y. Ishii, and C.-Y. D. Lu, J. Phys.: Condens.
Matter 23, 235105 (2011).

115601-6