Soft Matter
PAPER

Cite this: Soft Matter, 2014, 10, 5289

Structural rheology of focal conic domains: astress-quench experiment†
Shuji Fujii,a Shigeyuki Komura*b and C.-Y. David Luc
We study the dynamics of focal conic domain (FCD) formation in a thermotropic smectic phase under shear
stress. It is known that increasing the shear stress induces a non-equilibrium phase transition from a smectic
phase with FCDs (SmAI) to another smectic phase (SmAII) in which the layers are oriented. By quenching theshear stress from the SmAII phase to the SmAI phase, we ﬁnd three characteristic modes in the FCD

Received 19th January 2014
Accepted 28th April 2014

formation process. The ﬁrst mode is attributed to the edge dislocation dynamics induced by climb
motions. The second mode results from FCD formation. The ﬁrst and second modes show slowing
down close to the smectic–nematic transition temperature, implying that the dynamics are dominated

DOI: 10.1039/c4sm00146j

by dislocation unbinding. The third mode originates from the alignment of FCDs which form oily streaks.

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Such an alignment occurs when the shear stress balances the line tension of the oily streaks.

1

Introduction

When shear ow is applied to a smectic phase, non-equilibrium
dislocations and textural defects such as focal conic domains(FCDs) are formed, which signicantly inuence the smectic
rheology.1–12 Defect-mediated rheology in the smectic phase is
an important issue in structural rheology, which deals with theow properties inuenced by meso-scale structures in so
matter.13
Defects and dislocations aﬀect both rheological responses
and dynamic phase behaviors of so matter.14 Fig. 1 shows adynamic orientation diagram of 4-n-octyl-40 -cyanobiphenyl(8CB) thermotropic liquid crystals obtained by the present
authors.9 The smectic phase is divided into two types; the SmAI
phase in which the sample is occupied by micrometer-scale
FCDs, and the SmAII phase characterized by perpendicularly
oriented smectic layers.9,15,16 At constant temperature, the SmAI
phase converts to the SmAII phase by increasing the shear stresss. Shear stress induces a layer orientation transition, which is
not observed at equilibrium.15,16 In the vicinity of the equilibrium smectic–nematic transition temperature TSN, the SmAII
phase can be easily formed by the shear stress, and decreasing
the temperature expands the region of the SmAI phase.

Appearance of the SmAII phase and the increased FCD sizearound TSN are intimately related through the dislocation
unbinding mechanism.9,17 Shear stress dependence of the FCD
size and layer orientation in the dynamic phase diagram reect
the observed shear-thinning behavior.9
Although various non-equilibrium phases and structuraltransitions in so matter have been extensively investigated,2,18,19 the relationship between the structural transitionaDepartment of Chemistry, Nagaoka University of Technology, Nagaoka 940-2188,
Japan. E-mail: sfujii@mst.nagaokaut.ac.jpbDepartment of Chemistry, Tokyo Metropolitan University, Tokyo 192-0397, Japan.
E-mail: komura@tmu.ac.jpcDepartment of Chemistry, National Taiwan University, Taipei 106, Taiwan. E-mail:cydlu@ntu.edu.tw† Electronic supplementary information (ESI) available: Stress quench dataobtained by a cone-plate shear cell made of stainless steel and a parallel plateshear cell made of quartz are compared. Spatio-temporal images used for theconstruction of Fig. 6 and 7 are also shown. See DOI: 10.1039/c4sm00146j

This journal is © The Royal Society of Chemistry 2014

Dynamic orientation diagram of 8CB under shear stress as afunction of temperature T and applied shear stress s, reproduced fromref. 9. “SmAI” and “SmAII” denote the smectic phases with FCDs andperpendicularly oriented smectic layers without FCD, respectively, and“N” indicates the nematic phase. The vertical dashed line indicates thesmectic–nematic transition temperature TSN ¼ 33.4  C.
Fig. 1

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Soft Matterand the rheological behavior is inadequately understood.
Especially, the kinetic pathway between diﬀerent non-equilibrium phases remains unsolved because few experimental data
have been published.20,21 In this paper, we perform a stressquench experiment to elucidate the non-equilibrium FCD
formation in a thermotropic smectic phase. Specically, weinvestigate the temporal mechanical response of the dynamicaltransition during sudden quenching of the shear stress fromthe SmAII phase to the SmAI phase.
In lyotropic systems, it is known that multilamellar vesicles(onions) are formed by applying a shear ow to a lamellar
phase, and the onion size depends on the shear rate.22 Recently,
one of the authors studied the growth of the onion structure bya shear-quench experiment.23,24 We found that the transition
pathway can be either continuous or discontinuous dependingon the quench depth of the shear rate. During discontinuoustransitions, onions dissociate into planar lamellae in two steps,
and new onions of diﬀerent sizes are formed. Notably, FCDs
and onions belong to the same class of defect, but their
Gaussian curvatures diﬀer.25 In fact, the elasticity of FCDs andonions depends on the eﬀective surface tension of the layered
systems.10 Hence, the non-equilibrium transitions of FCDs andonions are expected to share some similar features.
Section 2 of this paper describes our shear stress-quench
method and tested materials. In Section 3, we explain our
experimental results for diﬀerent temperatures, and show that
FCD formation is characterized by three distinctive dynamical
modes. Possible physical origins of these modes are proposed
in Section 4. Section 5 summarizes our ndings.

2 Experiment
We used the thermotropic liquid crystal 4-n-octyl-40 -cyanobiphenyl (8CB). 8CB was purchased from Synthon Chemicals
GmbH Co., Germany and was used without further purication.
The equilibrium phase sequence and the rheological property
of 8CB have been well characterized in the literature.9,10,15,16
Rheological measurements were performed using a stresscontrolled rheometer, Anton Paar MCR-300, equipped with atruncated cone-plate (made of stainless steel). The diameter andthe cone angle were 50 mm and 0.017 rad, respectively.
Microscopy observation under shear ow was performed
using a polarized microscope attached to another rheometer,
Anton Paar MCR-501, with a parallel plate (made of quartz). The
diameter of the quartz shear cell was 50 mm, and the sample
thickness was xed to 150 mm. Incident light was polarized alongthe ow direction, and the analyzer was aligned perpendicular tothe ow direction. This setup enables simultaneous measurement of the viscosity and the structure under crossed-nicols. We
conrmed that the rheological data obtained by MCR-300 and
MCR-501 qualitatively coincide (see Fig. S1 of the ESI†).
During the structural observation, the ow patterns wererecorded as movies at a frame rate of 30 fps. The obtained
movies were converted into spatio-temporal images using
ImageJ. Here the optical intensity of a line of pixels along thevorticity direction (the le edge) of each frame was arrayed in atime-sequential order. Since the microscopy images were5290 | Soft Matter, 2014, 10, 5289–5295

Paperobtained under crossed-nicols, the optical intensity reects thebirefringence intensity. To evaluate the FCD density, the opticalintensity at each time in the spatio-temporal image was averaged over 200 mm along the vorticity direction. As described inFig. S2 and S3 of the ESI,† the position of the FCD was denedby the focal point of an ellipse (apex). Time evolution of the FCD
positions in the vorticity direction was also extracted from thespatio-temporal images. These images directly reveal thedensity variation and the alignment behavior of FCDs.
Prior to each stress-quench experiment, the sample was presheared under an initial shear stress si ¼ 85 Pa for 200 s. This is
suﬃciently long for the system to reach a steady state, and toeliminate the ow history eﬀect generated by the sample
loading processes. Once the system reached the steady state, theshear stress was suddenly decreased from si ¼ 85 Pa to a nal
value sf, and the resultant shear rate g_ and the ow behavior
were observed as a function of time t. The stress-quench
experiment was repeated at diﬀerent temperatures. Note that, at
si ¼ 85 Pa, the system is in the SmAII phase over the temperature range T ¼ 31.5–33.2  C (see Fig. 1). Below T ¼ 31.5  C and s< 85 Pa, the SmAI phase is lled with FCDs.

3 Results
Fig. 2(a) and (b) show time sequences of the snap shots aer
quenching the system from si ¼ 85 Pa to sf ¼ 0.1 Pa at T ¼
33.0  C and from si ¼ 85 to sf ¼ 1.5 Pa at T ¼ 31.0  C. The
brightness of the images reects the birefringence intensity.
The rst image in Fig. 2(a) (at t ¼ 0 s) is considerably brightbecause the smectic layer adopted a perpendicular orientation.9,15,16 Aer the stress-quench, the optical intensity rapidly
decreased within a few seconds. The relaxation of the intensity
is caused by a ip of smectic layers from the perpendicular toparallel orientation. At approximately t ¼ 50 s, FCDs appeared
as parabolic structures,26,27 and their number density increased
over time. By contrast, the rst image (at t ¼ 0 s) in Fig. 2(b)reveals an inhomogeneous birefringence pattern imposed by
FCDs. At later times, one can see parabolic patterns as well as
circular objects, which are also FCDs with smaller sizes. The
FCD size distribution changed over time as the population oflarger FCDs increased. At approximately t ¼ 900 s aer the
stress-quench, large FCDs aligned along the ow direction (oilystreaks) appeared. The FCDs were smaller at T ¼ 31.0  C than at
T ¼ 33.0  C, which is consistent with our previous result.10
Fig. 3 plots the measured shear rate g_ as a function of time t
for stress-quenches at diﬀerent temperatures (a) T ¼ 33.0, (b)32.0, (c) 31.0 and (d) 30.0  C. The rst mode, the initial decay of_ can be tted by a single exponential function as explainedg,below. When the system underwent the non-equilibrium phasetransition, the shear rate exhibited the second decay mode,
which accompanies FCD formation. As the nal stress sf
decreased, the fraction of the second mode markedly increased.
Interestingly, when the system was quenched within the SmAI
phase at T ¼ 31.0 and 30.0  C (Fig. 3(c) and (d), respectively), theshear rate showed anomalous behavior depending on the
stress-quench depth. As the quench depth increased, the two
decay modes appeared, similar to those at T ¼ 33.0 and 32.0  CThis journal is © The Royal Society of Chemistry 2014

PaperSoft Matter

Fig. 2 Time sequence of polarized microscope images after quenching the system at (a) T ¼ 33.0  C and (b) T ¼ 31.0  C. Shear stress was
quenched from si ¼ 85 Pa to sf ¼ 0.1 Pa in (a) and from si ¼ 85 Pa to sf ¼ 1.5 Pa in (b), respectively. The horizontal direction is the ﬂow direction.
The size of each image is 150 mm  200 mm (the scale bars correspond to 50 mm). The brightness of these images reﬂects the birefringenceintensity.

(Fig. 3(a) and (b), respectively). At lower temperatures, a thirdmode appeared during which the shear rate increased from its
minimum. The third mode became more apparent as thetemperature further decreased. In the following, we analyze
these three modes more quantitatively and discuss their physical origins.

Fig. 3 Log–log plots of the measured shear rate g_ as a function of the elapsed time t after the stress-quench from si ¼ 85 Pa to sf ¼ 60, 40, 20,
10, 7, 5, 3, 2, 1.5, 1 Pa (from upper to bottom) for (a) T ¼ 33.0, (b) 32.0, (c) 31.0 and (d) 30.0  C. The ﬁlled and the open symbols correspond to thestress-quench within the SmAII phase and from the SmAII to the SmAI phases, respectively. In (c) and (d), the stress-quench was applied within the
SmAI phase only. Solid curves are the ﬁts using a single exponential function with a characteristic time s1 [see eqn (1)].

This journal is © The Royal Society of Chemistry 2014

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3.1

Paper

Formation of dislocations: rst mode

As mentioned above, the rst decay mode can be empiricallytted by a single exponential function_ ¼ g_ 0 exp(t/s1),
g(t)

(1)

where s1 is the relaxation time. Fig. 4 plots s1 as a function of sf
at diﬀerent temperatures. We observe that s1 levels oﬀ andbecomes constant when sf < 1 Pa. On the other hand, s1
increases as the temperature approaches TSN.
Fig. 5 plots the temperature dependence of s1 at sf ¼ 1 Pa. As
TSN is approached from below, s1 increases in two steps. At s ¼
85 Pa, the crossover temperature coincides with the boundary
temperature (T x 31.3  C) between the SmAII and the SmAI
phases. For T > 31.3  C, the initial shear stress si ¼ 85 Pa is
located inside the SmAII region, while it is within the SmAI
phase for T < 31.3  C. The increase of s1 as a function of the
reduced temperature, dened as 3 ¼ (TSN  T)/T, obeys diﬀerent
power-law behaviors; s1  30.140.03 and s1  32.960.02, aboveand below T ¼ 31.3  C, respectively.
Since the smectic layers are perpendicularly oriented in the
SmAII phase, the ip of the layers from the perpendicular to theparallel orientation accompanies a bending deformation. Accumulated stress of smectic layers as a result of the bending
deformation can be relaxed by creation and proliferation of edge
dislocations.28 Under shear stress, these dislocations exhibit
climb motion,5–7,29 and reduce the shear rate by introducing theirline tension. Note that edge dislocations drive the FCD dynamics,
since they connect FCDs into oily streaks.30 Later, we will attribute the rst mode to the climb motion of edge dislocations.
3.2

Formation of FCDs: second mode

Fig. 6 shows the positions of FCDs extracted from the spatiotemporal images for the conditions described in Fig. 2. The

Fig. 4 Semi-log plot of s1 as a function of sf obtained from Fig. 3 for T¼ 33.0, 32.0, 31.0, 30.0  C. As indicated by the horizontal dashed lines,
s1 takes a constant value for small sf.

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Fig. 5 Plot of s1 at sf ¼ 1 Pa as a function of temperature T obtained
from Fig. 4. Solid curves show s1  30.140.03 (right) and s1  32.960.02(left), where 3 is the reduced temperature 3 ¼ (TSN  T)/TSN. The verticaldashed line indicates the smectic–nematic transition temperature TSN¼ 33.4  C.

spatio-temporal images under each quench condition are
shown in Fig. S2 of the ESI.† The positions of FCDs at very shorttimes are not shown because they were not easily determined in
the unfocused images. Comparing Fig. 6(a) and (b), we recognize that the FCD density depends strongly on the temperature;
FCDs are more dilute at a higher temperature and crowded at alower temperature. As time progresses, FCDs gradually alignalong the ow direction.
Fig. 7 shows how the optical intensity of spatio-temporal
images changes over time. These images were spatially averaged
in the vertical direction (see Section 2). Again, the opticalintensity reects the number density of FCDs, because it is
proportional to the birefringence intensity generated by FCDs.
The characteristic time s2 of the second mode was estimated bytting the optical intensity to the empirical formula:Fig. 6 Time evolution of FCD positions at (a) T ¼ 33.0  C and (b) T ¼
31.0  C. The position of each FCD was deﬁned by its focal point. The
stress-quench condition is that described in Fig. 2.

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PaperSoft Matter

Fig. 7 Plot of averaged optical intensity I as a function of time t at (a)T ¼ 33.0  C and (b) T ¼ 31.0  C. Solid curves are the ﬁts using a single
exponential function with a characteristic time s2 [see eqn (2)].

I(t) ¼ I0[1  exp(t/s2)].

(2)

According to the rheological measurement, the obtained s2
indeed corresponds to the second mode of the shear rate
relaxation. Hence the present mode can be attributed to theformation of FCDs and their evolution to the densely packedcongurations.
The temperature dependence of s2 is plotted in Fig. 8. The
shear stress dependence of s2 was not investigated because oflimited experimental data. We observed that s2 increases as TSN
is approached from below and obeyed a diﬀerent power-law
behavior; s2  31.030.02. Such slowing down behavior impliedthat FCD formation was inuenced by a dislocation unbinding
mechanism.17

Fig. 9 log–log plot of s3 as a function of sf for diﬀerent temperatures,
T ¼ 31.0, 30.5, 30.0  C.

3.3

Alignment of FCDs: third mode

The third mode manifests as an increase in the shear rate (seeFig. 3(c) and (d)). Comparing Fig. 3 with Figs 2 and 6, the thirdmode is temporally consistent with the FCD alignment. Shortlyaer the stress-quench t ¼ 100–200 s in Fig. 6(a), FCDs are not
yet aligned along the ow direction. The alignment of FCDs
occur at t ¼ 700–900 s, following the shear rate decay. A similarordering process is observed in Fig. 6(b). Hence, we attribute thethird mode to the FCD alignment, i.e., adjustment of oily
streaks along the ow direction. Incidentally, a similar
phenomenon was observed in the lane formation of colloidal
dispersions under an electric eld.31
We have characterized the third mode by s3 when the shear
rate took minimum values in Fig. 3. In Fig. 3(a) and (b),however, we did not determine s3 because there was nominimum in the shear rate. Fig. 9 plots s3 as a function of sf atthree diﬀerent temperatures. Clearly, the alignment of FCDs
depends on the nal shear stress sf. The third mode s3 significantly reduces as sf increases, leveling oﬀ at high sf. Since oily
streaks consisted of FCDs connected by edge dislocations,30 the
FCD alignment could be inuenced by the dislocation
unbinding. However, s3 did not change appreciably with thetemperature under our experimental conditions. Hence, edge
dislocations may not necessarily contribute to the FCDalignment.

4 Discussion

Fig. 8 Plot of s2 as a function of temperature T. The solid curve is s2 31.030.02. The vertical dashed line indicates the smectic–nematictransition temperature TSN ¼ 33.4  C.

This journal is © The Royal Society of Chemistry 2014

Three characteristic times observed in our stress-quench
experiment have been attributed to the climb of edge dislocations, the FCD formation, and the alignment of FCDs, respectively. The slowing down behavior of s1 and s2 in the vicinity of
TSN suggests that dislocation unbinding aﬀects the climb
motion and FCD formation.

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Paper

In Fig. 10, a possible layer ipping scenario is illustrated.
During stress-quenching from the SmAII to SmAI phases, thebirefringence dramatically reduces on a time scale comparable
to the rst mode, indicating that the layer ips from theperpendicular to parallel orientation. The initial perpendicularorientation becomes unstable under low stress. The resulting
large amplitude undulation induces a state like (b). The owthen encourages the layers to disassemble along the gradientdirection, as illustrated in (c). The layers can reconnect in theneutral (vorticity) direction to form edge dislocations as drawn
in the upper section of (d). Partially merged edge defects with
opposite Burgers vectors, as illustrated in the lower section of(d), become oily streaks.30 As shown in Fig. 4, s1 behaves similarly regardless when the initial state si corresponds to SmAI or
SmAII. Hence, the layer-ipping process (Fig. 10(a)–(c)) should
occur more rapidly than the defect climb process discussed
earlier.
As argued before, undulation instability leads to edge
dislocation when the smectic layer with parallel orientation
undergoes the dilation deformation under shear ow.32–35
Elementary edge dislocations may transform into FCDs to
reduce the core energy.26,28,30,32,36 Hence, the creation of edge
dislocations is essential for structure formation in the smectic
phase. Under shear stress, edge dislocations undergoing climb
motion inuence the shear rate depending on the mobility b of
the dislocation.27,37 Oswald and Kleman studied the collective
climb motion of edge dislocations under compression in
Grandjean-Cano wedge with an angle a.34 They argued that,when the smectic sample is slightly compressed, the stressrelaxes due to the cooperative climb motion of dislocations witha characteristic timesclimb x

d
:
Bab

(3)

Here d is the sample thickness and B is the layer compression
modulus. At high defect densities, d can be replaced by the FCD
size. Using typical values of d x 10 mm, B x 106 Pa,38,39 s ¼ 0.017
rad, and b x 108 m2 s kg1, we obtain sclimb x 0.1 s, which isroughly consistent with the experimental value s1 of the rst
mode (see Fig. 4).

Fig. 10 Proposed layer-ﬂipping scenario. Horizontal and vertical axes
are the directions of vorticity and velocity gradient, respectively. (a)Initially, layers are perpendicularly oriented in the SmAII phase. (b)Layers destabilize and undulate with large amplitude. (c) Layers
disassemble along the velocity gradient direction to allow for an easierﬂow. (d) Layers reconnect in the vorticity direction. The top section
shows well-merged edge dislocations, and the lower section illustrates a cross-section of an oily streak.

5294 | Soft Matter, 2014, 10, 5289–5295

One might also consider the contribution of the layer
undulation mode to the rst mode s1. The typical relaxation
time of the undulation mode is given by40sund x

h3
;
Kq2

(4)

where h3 is the in-plane viscosity, K the curvature elastic
modulus, and q the wavenumber of the undulation. Typicalvalues of h3 x 0.1 Pa s, K x 1011–1012 N,27,41 and q x 105 m1(corresponding to the FCD size) yields sund x 1 s, again
comparable to s1. As observed for s1, the undulation time sund isalso expected to increase as TSN is approached. However, typicalsund values are of the order of millisecond.42 This means that
evaluating q from the FCD size (10 mm) is not justied. Hence,
we attribute s1 to the edge climb motion rather than the layer
undulation mode.
As previously reported, FCDs are formed under shear ow.1,9Lu et al. theoretically explained the shear-thinning mechanismby the creation–annihilation dynamics of the dislocation loops.7
In our earlier study, we found that the SN-transition, which isdriven by dislocation unbinding, aﬀects the mechanical property. We reported that the average FCD size varies as L 
g_ 0.230.5.10 Although L has not been theoretically predicted,
Meyer et al. argued that the distance between edge dislocations
behaves as g_ 0.2.5,6 On the other hand, the unbinding transition theory17 predicts that the size of the dislocation loop
depends on the reduced temperature as 30.5. Hence, we
expect that the FCD size is determined by unbinding propertiesof the dislocation loop, which characterize s2. Very recently,
Chatterjee and Anna studied the eﬀects of shear and dilation
ows on toroidal and parabolic FCD formations.11,12 They
showed that the critical dilation required for parabolic FCD
formation increases under shear ow because such ow annihilates dislocations.
The smectic phase with FCDs exhibits the yield stress in theorder of sy x 102–100 Pa.1,9 The yield stress originates from theline tension of oily streaks that depends on its internal structure. According to Boltenhagen et al., the main contribution to
the line tension G is the layer compression energy, given by30


e
a
;
(5)
GxBd 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1  e2 dwhere e and a are eccentricity and major elliptic axis of a FCD,
respectively. Our FCD measurements yield e x 0.6 and a x 10–
100 mm. Using the previously mentioned values, we estimate
G x 102 Pa, which is roughly consistent with the experimentally observed yield stress.9 We showed that the characteristic
time s3 increases as sf approaches sy, whereby the shear rate
decreases to zero. Hence, we consider that s3 is determined by
the mechanical balance between the applied stress sf and theline tension of the oily streak G.

5 Conclusion
We studied the dynamics of FCD formation in the smectic
phase by performing stress-quench experiments on the thermotropic liquid crystal 8CB. In particular, we investigated bothThis journal is © The Royal Society of Chemistry 2014

Paperthe rheological response and the structural evolution as thesystem transforms from the SmAII phase under high shear
stress to the SmAI phase under low shear stress. We identied
three distinct characteristic times. The rst mode is attributed
to the climb motion of edge dislocations created when layers
ip from perpendicular to parallel orientations. The second
mode corresponds to the FCD formation. Both of these decay
modes also occur when the system is quenched within the SmAI
phase. The slowing down of the characteristic times s1 and s2 inthe vicinity of the SN-transition indicates that the dislocation
unbinding inuences the formation of edge dislocations. The
third mode originates from the alignment of FCDs into oily
streaks. Such a FCD alignment is determined by the balance
between the shear stress and the line tension of the oily streak.

Acknowledgements
Both SF and SK acknowledge support from the Grand-in-Aid for
Scientic Research (B), grant no. 25287107 from the Ministry of
Education, Culture, Sports, Science and Technology (MEXT) of
Japan. SF thanks T. Takahashi, H. Orihara and Anton Paar Co.
Ltd for the use of their rheometer and the microscopy devices.
SK would like to further acknowledge support from the Grantin-Aid for Scientic Research on Innovative Areas “Fluctuation& Structure” (grant no. 25103010), grant no. 24540439 from the
MEXT of Japan, and the JSPS Core-to-Core Program “International research network for non-equilibrium dynamics of so
matter”.

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