Materials 2014, 7, 5146-5168; doi:10.3390/ma7075146
OPEN ACCESS

materials
ISSN 1996-1944
www.mdpi.com/journal/materials
Review

Structural Rheology of the Smectic Phase
Shuji Fujii 1,2 *, Shigeyuki Komura 3 and Chun-Yi David Lu 4
1

Department of Chemistry, Nagaoka University of Technology, Nagaoka 940-2188, Japan
Department of Materials and Interfaces, Weizmann Institute of Science, Rehovot 76100, Israel
3
Department of Chemistry, Tokyo Metropolitan University, Tokyo 192-0397, Japan;
E-Mail: komura@tmu.ac.jp
4
Department of Chemistry, National Taiwan University, Taipei 106, Taiwan;
E-Mail: cydlu@ntu.edu.tw
2* Author to whom correspondence should be addressed; E-Mail: sfujii@mst.nagaokaut.ac.jp;
Tel.: +81-258-47-9302; Fax: +81-258-47-9300.
Received: 1 April 2014; in revised form: 13 June 2014 / Accepted: 24 June 2014 /Published: 16 July 2014

Abstract: In this review article, we discuss the rheological properties of the thermotropic
smectic liquid crystal 8CB with focal conic domains (FCDs) from the viewpoint of structuralrheology. It is known that the unbinding of the dislocation loops in the smectic phase drives
the smectic-nematic transition. Here we discuss how the unbinding of the dislocation loops
affects the evolution of the FCD size, linear and nonlinear rheological behaviors of thesmectic phase. By studying the FCD formation from the perpendicularly oriented smectic
layers, we also argue that dislocations play a key role in the structural development in layered
systems. Furthermore, similarities in the rheological behavior between the FCDs in thesmectic phase and the onion structures in the lyotropic lamellar phase suggest that thesesystems share a common physical origin for the elasticity.
Keywords: soft matter; structural rheology; smectic phase; dislocation loop; focal conic
domain; unbinding transition; lamellar phase; onion structure

1. Introduction
Rheology is a fundamental issue in soft matter science. One of the most successful achievementsin the rheology of soft matter is the Doi-Edwards model, which describes the viscoelastic response of

Materials 2014, 7

5147

entangled polymer melts [1]. This model guides further theories and experiments which contribute not
only to the industrial application of polymer materials but also to the progress of the basic polymer
science. In contrast to the success in the polymer systems, the rheology of soft matter with meso-scale
structures is still a developing field. Structured fluids such as foam, emulsions and colloidal systems
as well as polymers have been also widely studied for many industrial applications [2–5]. However, theattempt to describe their universal rheological properties has only started using the concept of “soft glassy
rheology” [5,6]. Besides these glassy materials, the rheology of surfactant systems which exhibit the
gyroid phase with a three-dimensional periodic structure, or the sponge phase with randomly connectedbicontinuous interface remains unexplored except for some pioneered studies [7–12]. Their uniqueviscoelastic responses arise predominantly from deformation of meso-scale internal structures whose
rearrangement can be easily induced under deformation or flow.
The rheology of soft matter looks for a fundamental understanding in terms of the micro/meso
structures of the systems. Representative examples of “structural rheology” include emulsions, foams,
colloidal dispersions, surfactant solutions, and liquid crystals. Molecular systems such as a lubricant
confined in a narrow space also exhibit various responses depending on their microstructure [13–15].
Since there are many different structures, to unify the rheology of structured fluids is an attractive yet
challenging subject. Furthermore, as the soft structures can also evolve in the flow, a good selection of
the structure unit is essential to understand the system. If we can establish a fundamental principle forthe “structural rheology”, the significance of the soft matter in the industrial application will increase. In
this review, we take defects as the key structures to understand the smectic rheology.
In this review article, we discuss the structural rheology of thermotropic smectic liquid crystal as a
typical example [16–19]. Among various soft materials which spontaneously form internal structures
such as lamellar, hexagonal, cubic, and gyroid phases [20–23], the simplest one-dimensional periodic
structure is the smectic-A phase in thermotropic liquid crystals. The smectic liquid crystals exhibit
a solid-like response in the layer perpendicular direction and a fluid-like response within the layers.
Although the solid and the fluid-like responses do not mix for the smectic with a perfect alignment,
self-organized textures combine these responses and give rise to the viscoelasticity. Even in such
simple systems, phenomena such as shear-thinning and the orientation transition of the smectic layer
are observed once a flow field is applied [24–29]. In lyotropic lamellar phases, it is known that bilayer
membranes form multi-lamellar vesicles (onions) under shear flow [30–34].
As Horn and Kleman [35] pointed out in their pioneering work, smectic rheology is influenced by
defects in the bulk. It has been explained that shear-thinning behavior depends on the defect dynamics
and/or the defect density [25,36–38]. Although the importance of defects in the smectic rheology is acommon understanding, there are relatively few studies that focus on the role of defects either explicitly
or systematically [39–42]. Meyer et al. and Lu et al. [36–38] studied the shear-thinning behavior byconsidering the dynamics of screw dislocations and dislocation loops. They found that the theoretically
predicted shear-thinning behavior γ̇ ∼ σ m , where γ̇ is the shear rate and σ the shear stress, was consistent
with the experimental results within a limited range of the shear rate.
The thermotropic smectic liquid crystal 8CB changes from the crystalline phase to the smectic phase
at T = 21.5 ◦ C, and further to the nematic phase at TSN = 33.4 ◦ C. In this review article, using 8CB
as a typical example of smectics, we summarize the structural observation, the linear and nonlinear

Materials 2014, 75148

rheological behavior of the smectic-A phase close to (but below) TSN . In the next section, we briefly
explain the defects in the smectic liquid crystal phase. The temperature and the shear rate dependencesof the defect size are discussed in Section 3. In Section 4, the nonlinear rheology of the smectic phase isinvestigated from the viewpoint of unbinding of dislocations, and summarize them in a dynamic phase
diagram. The physical origin of the elasticity of the smectic phase with defects is suggested in Section 5.
In the following section, we explain the dynamics of defect formation induced by a non-equilibriumphase transition in the smectic phase under flow. Finally, we mention the similarities between texturaldefects in the thermotropic smectic phase and the onion structures in the lyotropic lamellar phase.
In our study, we did not perform any surface treatment of the shear cell. The lack of the surface
anchoring may induce the misalignment of the smectic layers and lead to the nucleation of focal conic
domains (FCDs). However, in our experiment, reproducible results could be obtained by applying the
pre-shear even without any anchoring treatment.
2. Defect Structures in the Smectic Phase
A liquid crystal has fluidity of a liquid and elasticity of a crystal. It also contains defects which
locally break the translational symmetry and form reconnected layers [43]. Smectic-A phase with alayered structure is perturbed by two types of line defects; edge dislocations and screw dislocations
which are parallel and perpendicular to the layer surface, respectively. These line defects appear in pairs
with opposite signs so that two screw dislocations with opposite signs linked by edge dislocations forma dislocation loop. Since the dislocation loop locally compresses the layer thickness, an increase in
the dislocation loop density causes the accumulation of strain energy. Furthermore, FCDs are formed
to relax the compression deformation [44,45]. FCDs are visible under an optical microscope and havesizes from a few micrometers to a hundred micrometers.
As shown in Figure 1, there are two types of FCDs which can be distinguished by the sign of theGaussian curvature of the layers. The first type, FCD-I, has negative Gaussian curvature forming a
toroidal shape, and is frequently observed in thermotropic liquid crystals. Whereas the second type,
FCD-II, has a concentric sphere like onion structure which is observed only in lyotropic lamellar phases.
As presented in Figure 1b, FCD-Is are connected by edge dislocations to form “oily streaks” which arecommonly observed both in thermotropic and lyotropic systems [46].
Not only the edge dislocations but also the screw dislocations also affect the FCD structure. In the
FCDs, the layers are folded into Dupin cyclides with ellipse and hyperbola which can be identified as
disclinations. Kleman et al. [47] and Meyer et al. [48] experimentally observed that the interaction
between FCDs and dislocations creates kinks on disclinations. As a result of this interaction, the screw
dislocations will align along the kink. Unbinding of the dislocation loops will thus increase the kinkdensity and may affect the FCD shape.
It has been recognized that formation of dislocation loops plays an important role in thesmectic-nematic (SN) transition [49]. Helfrich [50] proposed that an increase in the dislocation loop
density destroys the smectic order, and the unbinding of dislocation loops drives the SN-transition.
In his model, the stability of dislocation loops is determined by its energy per unit length, i.e., line
tension. When the temperature is increased, the line tension decreases and becomes negative above

Materials 2014, 7

5149

T = TSN . This leads to the spontaneous nucleation and proliferation of dislocation loops, and the
decay of the smectic order is reflected by the temperature dependence of the layer compression modulus
B. Benzekri et al. [51,52] showed that B decreases according to a power-law behavior with a criticalexponent given by the Nelson-Toner model [53]. Using freeze-fracture transmission electron microscopy
technique, Moreau et al. [54] showed that dislocation loop size indeed increases in the vicinity of theSN-transition for a lyotropic liquid crystal.
Figure 1. Schematic diagrams of (a) two different types of focal conic domains, FCD-I
and FCD-II; and (b) oily streak structure in which five vertical cross sections of FCD-I areconnected by four edge dislocations [45,46]. In the FCDs, the layers are folded into Dupin
cyclides with ellipse and hyperbola so that the curved layers remain equidistant. Reproducedby permission of EDP sciences.

(a)

(b)

5
4

3

FCD-I

2

FCD-II1

3. FCDs under Shear Flow
As Horn and Kleman presented [35], the FCD density increases by applying a shear flow. Theirexperimental observations suggest that non-equilibrium textural defects is additionally induced by theshear flow. Hence the smectic structures are strongly affected by the formation of non-equilibrium
defects. In this section, we discuss the relation between the proliferation of dislocation loops and the
FCD size and the shear rate [16].
Figure 2 shows polarized light microscope images of 8CB in the smectic phase under shear flow.
These images were obtained immediately after applying different shear rates for 10 min for various
temperatures ranging from the room temperature to TSN . The vertical and the depth directions correspondto the flow and the velocity gradient directions, respectively. It can be seen that strings of FCDs along
the flow direction fill the space. We also note that the FCD size decreases with increasing the shear
rate at each temperature. Under constant shear rate, on the other hand, the FCD size increases as thetemperature approaches TSN .

Materials 2014, 7

5150

Figure 2. Polarized microscope images of the smectic phase under shear flow at
different temperatures (a) T = 33.0; (b) 31.0; (c) 29.0; (d) 25.0 ◦ C and shear ratesγ̇ = 0.1, 1, 10, 20, 100, 200 s−1 . The shear flow was applied along the vertical direction of
the images which were taken just after the cessation of it. The scale bar corresponds to
100 µm.

To systematically study the effects of the temperature and shear rate on the FCD size, it is necessary to
determine the mean diameter L of FCDs. A microscope image taken under the shear rate γ̇ = 0.1 s−1 ispresented in Figure 3 as an example in order to show how to obtain L. As shown in the figure, the averageof L can be estimated by tracing the distinguishable outlines of the FCDs with circles. Some FCDs are
not used because their boundary contrast is too low. These FCDs are located either above or below thefocus plane. We calculate L as the number average within the focus plane region. Close to the transition
temperature TSN , one expects that the correlation length of the fluctuation and the dislocation size behave
similarly to obey the scaling law in terms of the reduced temperature t = (TSN − T )/TSN [55,56]. From
our experiment, we found that L depends on t and γ̇ as L ∼ γ̇ −0.2 (at each temperature) and L ∼ t−0.5(at each shear rate), respectively. As presented in Figure 4, the FCD size L can be scaled by the combined
variable γ̇ −0.2 t−0.5 .
Although there is no theory on the shear rate and/or temperature dependence of L, an analogous
scaling relation with the same exponent was predicted for the defect spacing which varies as∼ γ̇ −0.2 [36,37]. Moreover, the average dislocation loop size diverges as ∼ t−0.5 according to thedefect model by Helfrich [50]. We remind that dislocation loops are formed by pairs of edge and screw
dislocations, and FCDs are linked by edge dislocations. Since the accumulation of the strain energy
due to an increase in the defect density is the driving force for the FCD formation, the agreement ofthese exponents (0.2 and 0.5) suggests that the proliferation of dislocation loops controls the FCD size.

Materials 2014, 7

5151

Incidentally, it has been suggested that the non-equilibrium structural transition of lyotropic lamellar
phase under shear is governed by the dislocation size [29].
Figure 3. Typical microscope image used to estimate the FCD size at T = 25.0 ◦ C andγ̇ = 0.1 s−1 . The averaged diameter L of FCDs was estimated by tracing each FCD with a
red circle as shown in the picture. The scale bar corresponds to 50 µm.

Figure 4. Log-log plot of the FCD size L as a function of the shear rate γ̇ and the reduced
temperature t. Different symbols correspond to different temperatures. The scaling variable
is chosen to be γ̇ −0.2 t−0.5 so that all the data points fall onto a straight dashed line whose
slope is unity.

L [µm]

10102

33.0 ◦ C
31.0
29.0
25.0

slope = 1

1

0

10 0
10

1

10
0.2

γ̇ −0.2 t−0.5 [s ]Materials 2014, 75152

4. Nonlinear Rheology of the Smectic Phase
It is known that the smectic phase exhibits both shear-thinning behavior and yieldstress [24–26,28,35]. Although it has been anticipated that such rheological behavior is influenced by
defects [41,42], we have further shown in the previous section that these structures are FCDs which
originate from dislocation loops. In this section, we discuss how the presence of FCDs are reflected in
the rheology of the smectic phase [16].
Figure 5 shows the flow curves in the temperature range from 25.0 ◦ C to 34.0 ◦ C across theSN-transition temperature TSN = 33.4 ◦ C. These flow curves are obtained by measuring the steady-statevalue of the shear rate γ̇ when various values of the shear stress σ are applied. Within the power-law
behavior γ̇ ∼ σ m , m > 1 and m = 1 correspond to the shear-thinning and the Newtonian behaviors,
respectively. Since m > 1 for T < TSN , the smectic phase exhibits a shear-thinning behavior.
Figure 5. Log-log plot of the steady-steady shear rate γ̇ as a function of the applied
shear stress σ at different temperatures. The inset shows a typical curve (T = 33.1 ◦ C)
divided into three regimes. Regime 0 is fitted by the Herschel-Bulkley model given by
Equation (1); Regime I is fitted by the power-law behavior given by Equation (2); andRegime II corresponds to the Newtonian behavior. At the threshold stress σt , the transition
from Regime I to Regime II takes place. The SN-transition temperature TSN is 33.4 ◦ C.
4

10

10
10

3

10

10
10

2

γ̇ [s −1]

10

10
10

3
2

II

I

1
0
−1

0

−2
−1

10

σt
0

10

1

10

2

10

3 4 . 0 ◦C
33. 5
3 3 . 4 ( T S N)
33. 3
33. 2
33. 1
33. 0
32. 9
32. 8
32. 5
32. 0
31. 0
29. 0
27. 0
25. 0

1

10

0

10

−1

10

−2

10 −2
10

−1

10

10

0

1

10

2

10

3

10

σ [Pa]It should be noted that the flow curves are not fully described by a single power-law since the slope
in Figure 5 gradually changes as a function of the shear stress. Colby et al. [24] also observed a similar
behavior of the flow curve. In order to discuss the nonlinear rheological response in more detail, we
focus on the flow curve at T = 25.0 ◦ C as a typical example. First we realize that this flow curve
reaches to a finite stress value when the shear rate is extrapolated to zero. This means that there is a yieldMaterials 2014, 75153

stress σy , below which the flow ceases. Previously, Horn and Kleman [35] or Colby et al. [25] reported
that the smectic phase shows the yield stress. The flow curves for other temperatures also exhibit yieldstress; it decreases at higher temperatures and vanishes at TSN . Furthermore, focusing on the high-shear
flow curves near the transition point (e.g., T = 33.0 ◦ C), we see that the rheological behavior changes
from shear-thinning to Newtonian at a specific shear stress. The corresponding threshold stress value
σt required to become Newtonian shifts toward the lower value when approaching TSN . In the nematic
phase at higher temperatures, only Newtonian behavior is observed.
To extract the temperature dependence of the power-law exponent and the yield stress, the flow curve
was divided into three regions: shear-thinning region (Regime 0) showing the yield stress, shear-thinning
region (Regime I) described by a power-law, and Newtonian region (Regime II). Notice that σt represents
the boundary value between Regime I and Regime II. To estimate the yield stress in Regime 0, we use
the empirical Herschel-Bulkley (HB) modelσ = σy + Aγ̇ n(1)where A and n are parameters, and σy is the yield stress. This model has been frequently used to describe
the non-Newtonian behaviors of yield stress fluids [57–59]. In addition to the HB model for Regime 0,
Regime I was fitted with the power-law:
σ = C γ̇ 1/m(2)
Various quantities can be obtained by fitting these equations to the flow curve (see the inset of Figure 5).
The temperature dependence of each parameter is summarized in Figure 6. Whereas σy and A rapidly
decrease and vanish at TSN , the exponent n does not show a simple temperature dependence. On the other
hand, C and m are almost constant up to around T = 32.0 ◦ C and show a significant increase as TSN isapproached from below. The abrupt increase of C and m near TSN indicates that the enhanced criticalfluctuation or the proliferation of dislocation loops significantly affects the shear-thinning behavior. The
shear-thinning exponent m ≈ 1.7 obtained at the low-temperature region coincides with the theoretically
predicted exponent m = 5/3 by Kleman et al. [36,37].
As discussed in the previous section, the growth of dislocation loops causes the increase of the FCD
size. Horn and Kleman employed a dimensional argument to relate the yield stress σy and the FCD size
L by σy ∼ K/L2 , where K is the bending modulus of the smectic phase [35]. Notice that the value of K
is only weakly dependent on the temperature [60]. According to this relation, we see that a decrease ofσy near TSN corresponds to a rapid increase of L. Qualitatively, this temperature dependence of L agrees
with the direct observations of FCDs in Figures 2 and 4. Hence we expect that the growth of dislocation
loops influences the temperature dependence of σy through the increase of the FCD size. We shall further
discuss the elasticity of FCDs in the next section.
Next we discuss the transition from shear-thinning (Regime I) to Newtonian behavior (Regime II)using the dynamic phase diagram. In addition to σt obtained from Figure 5, the temperature dependenceof the viscosity η under constant shear stress was measured to construct non-equilibrium phase diagram.
The temperature dependence of η measured at shear stress of σ = 10, 30, 50, and 100 Pa is shown in
Figure 7. Several results are found; (i) at low temperatures, η decreases as the temperature increases;(ii) above a certain temperature T1 , η is almost constant; and (iii) at T2 slightly lower than TSN , a peakMaterials 2014, 75154

is observed. The two characteristic temperatures T1 and T2 shift toward lower values as the shear stressis increased.
Figure 6. (a) The yield stress σy , the pre-factor A, the exponent n in Equation (1), and (b) the
pre-factor C, the shear-thinning exponent m in Equation (2) as a function of the temperature
T . These values are obtained by the best fit in Figure 5 for each regime. Vertical dashed line
indicates the SN-transition temperature; TSN = 33.4 ◦ C.

4

C

1
0.5
0
25
4

27

29

31

33

27

29

31

33

2
0
25
3

35

2
0
25
1

n

(b)

m

A

σ y [Pa]

(a)

29

27

29

31

33

35

31

33

35

2
1
25

35

27

T [ C]
◦

0.5
0
25

27

29

31

33

35

T [ C]◦log η + a

Figure 7. Shear viscosity η as a function of the temperature T obtained at different applied
shear stress σ = 10, 30, 50, 100 Pa. Here log η is shifted by a constant a in order to have abetter visibility. Vertical dashed line indicates the SN-transition temperature; TSN = 33.4 ◦ C.

1

10 Pa (a=0.6)(a=0.4)30(a=0.2)50100 (a=0)

0

−1

−2
30

T1
31

T2
33

32

T [ C]◦

34

35

Materials 2014, 75155

The above results are summarized in the dynamic phase diagram presented in Figure 8. Here wefind that the shear stress dependence of T1 and the temperature dependence of σt are almost identicalto each other. Furthermore, both T1 and T2 change linearly with respect to σ. The temperature
estimated by extrapolating T1 and T2 to zero shear stress coincides with TSN at quiescent state, andthe phase diagram can be divided into three regimes. Comparing with the flow curves of Figure 5
and the temperature dependence of η in Figure 7, we see that the low-temperature region (T ≤ T1 )corresponds to the shear-thinning region (Regime I), while the temperature region T1 ≤ T ≤ T2 exhibits
the Newtonian behavior (Regime II). For convenience, these two regions are denoted by “SmAI phase”
and “SmAII phase”, respectively.
Figure 8. Dynamic phase diagram of 8CB under shear plotted against the temperature T
and the applied shear stress σ. The smectic becomes Newtonian above σt , whereas T1 andT2 are characteristic temperatures identified in Figure 7. The vertical dashed line indicates
T = TSN . The lines of σt and T1 coincide with each other. “SmAI ” and “SmAII ” denote thesmectic phases in Regime I and II, respectively, while “N” indicates the nematic phase.

100

σ[Pa]

80
60

SmA I

SmA I I

N

40
20
0
30

σt
T1
T2
31

32
T [ ◦ C]

33

34

Previously, dynamic orientation diagram of the smectic phase was constructed with the use ofrheo-physical methods such as small angle X-ray scattering under shear flow by Safinya et al. [27] andPanizza et al. [26], or rheo-dielectric measurement by Negita et al. [61]. Our dynamic phase diagram
qualitatively agrees with their results. Although Safinya et al. [27] and Negita et al. [61] presented theirdiagrams as a function of the temperature and the shear rate, we also confirmed that our diagram roughly
coincides with them by mapping the shear stress to the shear rate using the flow curves in Figure 5. As
shown in Figure 9, it is known that two orientation states are possible in the smectic phase: perpendicularorientation in which the layer normal is perpendicular to both the velocity gradient and flow direction,and parallel orientation in which it is parallel to the velocity gradient direction [62,63].

Materials 2014, 7

5156

Since the dynamic phase diagram obtained in Figure 8 and the orientation diagrams based onrheo-physical methods coincide, SmAI phase is a mixture of the perpendicular and parallel orientations,or a leak structure for which layers are cylindrically rounded [26]. On the other hand, SmAII phaseconsists of the perpendicular orientation. From this finding, we realize that the rheological behavior
and the layer orientation are closely linked to each other. Furthermore, since T2 coincides with TSN
for sufficiently low-shear stress (σ = 10 Pa), we anticipate that the peak in η reflects the precession
motion of the monomers in the SN-transition [27,61]. The shift of T2 toward lower temperatures
when subjected to high-shear stress implies that the SN-transition is induced by the shear flow.
In addition to the proliferation of dislocation loops in thermal equilibrium state, the shear-induced
SN-transition is due to the creation of non-equilibrium dislocation loops caused by the shear flow. The
shear-induced SN-transition originates from the unbinding of dislocation loops which are created bothunder equilibrium and out-of-equilibrium conditions. A rich rheological behavior associated with theshear-induced layer orientation has been also found in the lyotropic lamellar phases [64–66].
Figure 9. Schematic diagram of a smectic phase with perpendicular (left) and parallel(right) orientations under shear flow. ∇, v and n correspond to the flow gradient, flow
and vorticity directions, respectively. The SmAI phase consists of both perpendicular andparallel orientations, while only the perpendicular orientation appears in the SmAII phase.

Шnv(a) perpendicular(b) parallel

5. Linear Viscoelasticity of the Smectic Phase
In this section, we discuss the influence of FCDs on the linear viscoelasticity of the smectic phase [17].
We also argue the physical origin of the elasticity of the smectic phase with FCDs. As described in
the previous section, the dynamical smectic phase changes from the SmAI phase to SmAII phase as afunction of the shear stress and temperature. Since the SmAII phase exhibits only Newtonian behavior,
we mainly concentrate on the viscoelasticity of the SmAI phase.
In rheological measurements, the shear modulus G can be obtained by the ratio between the shear
stress σ and the strain γ as G = σ/γ. On the other hand, σ is given by the product of the viscosityη and the shear rate γ̇, i.e., σ = η γ̇. Viscoelastic materials exhibit both elastic and viscous responses
which can be measured by applying oscillating strain with an angular frequency ω and an amplitude of
γ0 ; γ = γ0 sin(ωt). The dynamic storage modulus G0 and the loss modulus G00 are determined by thefollowing relation:
σ = γ0 [G0 (ω) sin(ωt) + G00 (ω) cos(ωt)](3)

Materials 2014, 75157

In our experiment, all of the measurements were performed within the linear viscoelastic region which
was confirmed by the strain sweep tests.
Figure 10 shows the frequency dependence of G0 and G00 measured after the system is subjected toa given pre-shear stress. For all temperatures, G0 is always larger than G00 . Moreover, a plateau region
is observed in the low-frequency range, as Colby et al. also reported [24]. This solid-like viscoelastic
behavior is strongly correlated with the defect density. In fact, Larson et al. [28] showed that both G0
and G00 decrease when the defects are removed by applying a large amplitude oscillatory shear. Hencethe plateau modulus reflects the defect density.
Figure 10. Log-log plot of the dynamic storage modulus G0 and loss modulus G00 as afunction of the frequency ω at different temperatures. Filled and open symbols correspond
to G0 and G00 , respectively. Different symbols shown in (d) represent the applied pre-shear
stresses for which the steady states are obtained.

(a) T =33.0 ◦ C

3

10G ′ , G ′′ [Pa]G ′ , G ′′ [Pa]

3
2

10

1

10

0

10

−1

10 −2
−1
0
10 10
10

10

1

10

2

10

1

10

0

10

−1

10 −2
−1
0
10 10
10

2

10

ω [s −1](c) T =29.0 ◦ C

3

10

2

10

1

10

0

10

−1

10 −2
−1
0
10 10
10

10

ω [s −1]

1

2

10

10

ω [s −1]
G ′ , G ′′ [Pa]G ′ , G ′′ [Pa]3(b) T =31.0 ◦ C

1

2

10

(d) T =25.0 ◦ C

10

2

10

1

100 Pa

10

20

0

10

10
2

−1

10 −2
−1
0
10 10
10

1

10

2

10

ω [s −1]

Concerning the pre-shear stress dependence of the plateau value of G0 (denoted as G0 ) at different
temperatures, G0 becomes larger with increasing the pre-shear stress. However, as the temperatureapproaches to TSN such as at T = 33.0 ◦ C, it decreases at high pre-shear stress and deviates from a
simple scaling suggested in Figure 11. Comparing the pre-shear stress dependence of G0 to Figures 5
and 8, we notice that the shear stress value where G0 decreases is located near the boundary between
the SmAI and the SmAII phases. Thus, the plateau value of G0 reflects the defects associated withorientations of the layers. As discussed before, FCDs fill the SmAI phase under shear flow, whereas

Materials 2014, 7

5158

they are not observed in the SmAII phase. Hence, FCDs dominate the elasticity of the SmAI phase.
Once the orientation transition of the smectic layers takes place to become SmAII , the elasticity due to
FCDs vanishes.
Figure 11. Log-log plot of the plateau shear modulus G0 as a function of the shear rate γ̇
and the reduced temperature t. Different symbols correspond to different temperatures. The
scaling variable is chosen as γ̇ 0.2 t0.7 so that most of the data points fall onto a straight dashedline whose slope is unity.

G 0 [Pa]

10

10

10

3

33.2 ◦ C
33.0
32.0
slope = 1
31.0
29.0
27.0
25.0

2

1

−2

−1

10

γ̇

10
0.2 0.7

t

0

[s

10
−0.2

]

Similar to the temperature dependence of the yield stress, G0 also decreases as TSN is approached.
Since the FCD size L influences the shear modulus G0 , a similar scaling behavior found in Figure 4 is
expected to hold. Here the value of G0 at ω = 0.1 s−1 was chosen as G0 which is plotted in Figure 11 as afunction of the combined variable of γ̇ and t. Adopting the result of Figure 5, the measured steady shear
rate γ̇ for each applied pre-shear stress σ can be used for the scaling plot. To obtain the scaling plot for
G0 , we first determined the power-law dependence G0 ∼ γ̇ 0.2 at each temperature. Then the power-law
behavior of G0 as a function of t with an exponent 0.7 was extracted so that all the data points fall onto astraight line with a slope of unity, i.e., G0 ∼ γ̇ 0.2 t0.7 . Except for the data close to the border between the
SmAI and SmAII phases, all G0 values fall on a straight line. Below, we discuss the physical meaning ofthese scaling behaviors for G0 and L.
When comparing the two scaling relations G0 ∼ γ̇ 0.2 t0.7 and L ∼ γ̇ −0.2 t−0.5 obtained fromindependent measurements, it appears that G0 is almost inversely proportional to L, G0 ∼ 1/L, although
the temperature exponent is slightly different. In order to satisfy this relation, the proportionality
coefficient on the right hand side must have the dimension of surface tension, i.e., energy per unit area.
For layered systems such as the smectic phase or the lamellar phase, de Gennes and van der Linden√
proposed an effective surface tension given by γeff ' KB, where K and B are the bending and theMaterials 2014, 75159

compression moduli, respectively [55,67,68]. Here the numerical pre-factor is dropped. Thus the plateau
shear modulus of the smectic phase G0 should obey the following relation:
√
0 KB
(4)
G0 = C
L
where C 0 is the dimensionless proportionality coefficient. In the case of 8CB, it is known that K is
almost constant, K = (5.2 ± 0.3) × 10−12 N [60], whereas B decreases with increasing temperature
close to TSN . According to the experimental result of Benzekri et al., the temperature dependence of B
is given by B = (7.5 × 107 ) × t0.4±0.03 Pa [51,52]. To verify the validity of Equation (4), we compare√
the value of G0 L obtained from our experimental result and γeff = KB estimated from the literature,√i.e., G0 L = (4.56 × 10−3 ) × t0.2 Nm−1 and KB = (1.97 × 10−2 ) × t0.2 Nm−1 . It is remarkable thatboth quantities scale as ∼ t0.2 . This result implies that the temperature dependencies of G0 and L arerelated through the scaling B ∼ t0.4 .
Based on our experimental result, we conclude that the physical origin of the elasticity in the smectic√
phase is the effective surface tension γeff ' KB of the FCDs. An analogous picture also holds forthe elasticity of concentrated emulsions [69,70]. Furthermore, a similar relation to Equation (4) has also
been observed for the onion phase in surfactant solutions which can be identified with FCD-II [71,72].
The proportionality coefficient C 0 for the onion phase is about C 0 ≈ 0.4 – 1.2, which is fairly close toC 0 = 0.456/1.97 ≈ 0.23 obtained for the FCDs. The relatively small value of C 0 for 8CB may be dueto the polydispersity of the FCD size. We mentioned before that FCD-I can be observed not only in thethermotropic smectic phase but also in the lyotropic lamellar phase, while FCD-II appears only in thelyotropic systems. Formation of these textures depend both on the bending and the Gaussian moduli. As
shown in Figure 1, the main geometrical difference between FCD-I with toroidal shape and FCD-II withspherical shape is the sign of the Gaussian curvature. Therefore, the energy cost for the deformation of√
FCDs, as determined by KB, dominates the elasticity in these two systems in spite of the geometricaldifference. We expect that the origin of elasticity is a universal feature that is common to different
layered systems.
Equation (4) for the elastic modulus G0 is different from the relation for the yield stress σy ∼ K/L2
predicted by Horn and Kleman [35]. It should be noted, however, that G0 in our case was measured
within a linear regime, whereas the non-linear effect cannot be ignored in the estimation of yield stress.
Generally, the shear modulus and the yield stress are not proportional to each other.
6. FCD Formation Induced by Shear Quench
In the previous sections, we mentioned that the FCDs induced by the unbinding of dislocations
influence the viscoelasticity of the smectic phase. In this section, we explain our results in the
non-equilibrium FCD formation when the system is subjected to shear stress-quench from high to low
values [18]. Some studies on the FCD formation behavior under shear have been also reported [73,74].
Microscope images in Figure 12a,b show the time sequences of snap shots after quenching the systemfrom σ = 85 to 0.1 Pa at T = 33.0 ◦ C, and from σ = 85 to 1.5 Pa at T = 31.0 ◦ C, respectively.
Comparing with the dynamic phase diagram in Figure 8, one sees that conditions of the stress-quench inMaterials 2014, 7

5160(a) correspond to the FCD formation starting from the SmAII phase, while (b) is the FCD growth inside
the SmAI phase. The brightness in the image represents the birefringence intensity.
Figure 12. Time sequence of polarized images after quenching the system at (a) T = 33.0 ◦ C
and (b) T = 31.0 ◦ C. Shear stress was quenched (a) from σ = 85 Pa to 0.1 Pa, and (b) fromσ = 85 Pa to 1.5 Pa, respectively. The horizontal direction is the flow direction. The
FCDs are moving under the shear flow, and these images were picked out from the movies.
The size of each image is 150 µm × 200 µm. The brightness of the image reflects thebirefringence intensity. The scale bars correspond to 50 µm.

(a) T =33.0 ◦ C

t=0 s

50 s

100 s

200 s

400 s

900 s

100 s

200 s

400 s

900 s

(b) T =31.0 C
◦

t=0 s

50 s

The microscope image at t = 0 s in Figure 12a is considerably bright because of the perpendicularorientation of the smectic layers [16,26,27]. After the stress-quench, the birefringence intensity quickly
decays within a few seconds. Such a relaxation of the birefringence is caused by a flip of smectic layers
from the perpendicular orientation to the parallel one. After the relaxation, a parabolic pattern appears
around t = 50 s, showing the appearance of FCDs whose number density increases as a function of time.
At t = 0 s in Figure 12b, on the other hand, the birefringence intensity is inhomogeneous. In later times,
we see parabolic patterns as well as circular objects which are also FCDs with smaller sizes. There is a
distribution of the FCD size, and the population of larger sizes increases with time. Around t = 900 s inboth cases, there are large FCDs aligned along the flow direction.
Figure 13 shows the measured shear rate γ̇ as a function of time t for stress-quenches at the same
temperatures as in Figure 12. In Figure 13a for T = 33.0 ◦ C, open symbols correspond to thestress-quench within the SmAII phase, while closed symbols indicate the stress-quench from the SmAII
phase to the SmAI phase. As drawn in the graph, the initial decay of γ̇ can be fitted by a single
exponential function which is called the first mode. On the other hand, when the system undergoes anon-equilibrium transition from the SmAII to SmAI phases, the shear rate exhibits a second decay mode
as observed in Figure 13. In comparison with the microscope observations, we notice that the second
mode appears when the formation of FCDs starts. As the terminal stress is decreased, the fraction of thesecond mode gradually increases.

Materials 2014, 7

5161

Figure 13. Log-log plot of the measured shear rate γ̇ as a function of the elapsed time t after
the stress-quench from σ = 85 Pa to 60, 40, 20, 10, 7, 5, 3, 2, 1.5, 1 Pa (from upper to bottom)
for (a) T = 33.0 ◦ C and (b) T = 31.0 ◦ C, respectively. Open and closed symbols correspondto the shear quench within the SmAII phase and those from the SmAII to the SmAI phases,
respectively. Data for (b) correspond to the stress-quench only within the SmAI phase. Solid
curves are the fits using a single exponential function with a characteristic time (first mode).

The obtained relaxation time for the first mode shows a slowing down at TSN . Since the smectic
layers in the SmAII phase consist of a perpendicular orientation, the relaxation of the layer orientationaccompanies a creation of edge dislocations before the FCD formation. It is known that the dislocations
exhibit climb motion under shear stress [75,76]. The characteristic time for the climb motion of edge
dislocation under stress is described bydτclimb ∼(5)Bαbwhere d is the sample thickness, α the angle of the cone-plate shear cell, and b the mobility of the edge
dislocation. Using typical values d ≈ 10 µm, B ≈ 106 Pa, α = 0.017 rad, and b ≈ 10−8 m2 s/kg, we
obtain τclimb ' 0.1 s, which roughly corresponds to the experimentally observed first mode. Hence thefirst relaxation can be attributed to the climb motion of edge dislocation, and the slowing down close to
TSN suggests that the climb motion is affected by the unbinding of the dislocation. Moreover, a similar
slowing down is observed for the second mode. The validity of Equation (5) can be further checked bysystematically changing d and α.
When the system is quenched within the SmAI phase at T = 31.0 ◦ C as shown in Figure 13b, the shear
rate shows anomalous behavior depending on the stress-quench depth. As the quench depth is increased,
the two decay modes are observed similar to those for T = 33.0 ◦ C. However, there is a distinct thirdmode in which the shear rate increases (rather than decreases) after the double relaxation. Since this thirdmode corresponds to the time region when the alignment of FCDs takes place, it can be attributed to theformation of oily streaks. The third mode becomes slower when the temperature is decreased. Noticethat oily streaks consist of FCDs connected by edge dislocations as depicted in Figure 1b. Hence theMaterials 2014, 7

5162

alignment of FCDs is affected by the dislocation unbinding [46]. Slowing down of these characteristic
times in the vicinity of the SN-transition indicates that the dislocation unbinding dominates not only the
SN-transition but also the dynamics of textural defects.
Here we shall qualitatively discuss the eccentricity of FCDs under shear flow. In Figure 12,
elliptic FCDs can be seen with minor axis aligned along the flow direction. The eccentricity of
the FCDs is caused by the shear since the FCDs at quiescent state are mostly circles as shown in
Figure 3. When the shear stress is quenched, the layers will first disassemble along the velocity gradientdirection, and may reconnect with the slightly tilted state to accommodate FCDs with the large Burger
vectors [18]. Asymmetry of the FCDs along the vorticity direction would reflect the tilted layers in
the FCDs. Moreover, the dynamic coupling between the dislocation loops and the shear flow may
induce the distortion of the FCDs since the dislocation loops can adapt to the applied shear stress [38].
Kleman et al. [47] and Meyer et al. [48] indeed pointed out that the interaction of the FCDs withdislocations causes the distortion of the FCD shape. Detailed analysis on the eccentricity would clarify
how the dislocation loops attribute to the FCD structure.
Finally, it is interesting to point out the similarity between the FCD (FCD-I) formation from theperpendicularly oriented layers and the onion (FCD-II) formation from the planar lamellae with parallelorientation. In our experiment, the double decay modes in the shear rate at fixed shear stress indicatethat the viscosity increases with two steps. Such a behavior is also observed in the shear-induced onion
formation at fixed shear rate [31,32]. Especially, the creation of dislocations prior to the FCD formation
coincidences with the previous observation in the lyotropic system in which the increase of the defectdensity is essential for the onion formation [33]. Hence the proliferation of the dislocations plays animportant role in the structural development from the planar layers for both thermotropic and lyotropic
cases. A qualitatively similar time evolution of the viscosity in these systems suggests that the structuraldevelopment dominated by defects is an universal mechanism in the layered systems.
7. Conclusions and Outlook
In this review article, we have discussed that the following properties are closely related to theunbinding of the dislocation loops; (i) scaling behavior of the FCD size; (ii) temperature dependenceof the yield stress; (iii) shear-induced SN-transition; (iv) physical origin of elasticity; and (v) FCD
formation from the SmAII phase. In particular, we have clarified the physical origin of the elasticity bycomparing the scaling relations for the FCD size and the shear modulus. This result indicates that thedefects significantly affect the smectic rheology. Furthermore, the similarity in the rheological properties
between FCDs and onions is a noteworthy consequence.
It should be noted that the origin of the nonlinear rheological behavior still remains to be clarified.
Both the yield stress and the plateau shear modulus, commonly used to characterize the elasticity, dependon the FCD size. The shear modulus originates from the effective surface tension, whereas the source ofthe yield stress is poorly understood. Nonetheless, the estimated yield stress using an empirical relation
σy ∼ K/L2 with L ≈ 10 –102 µm roughly coincides with our experimental observation in Figure 6.
More detailed study will shed light on the yield stress in the smectic phase. By the same token, theshear-induced onion formation in lyotropic systems is strongly concerned with interactions of defects

Materials 2014, 7

5163

such as dislocations and oily streaks [30,33]. Understanding the origin of the nonlinearity in the smectic
phase with defects will also lead to the elucidation of the shear-induced structural transition.
We expect that the concept of structural rheology is applicable not only to the smectic phase but also
to other structured fluids [2,3,5,77–79]. One of the most interesting systems, which could be tackled bya similar concept, is the rheology of the blue phase in cholesteric nematic liquid crystals. In this phase,the interaction between disclinations generates the yield stress, and anomalous rheological behavior is
expected depending on the type of disclination networks [80,81]. Especially, shear-induced breakup andreconnection of the disclination network may induce a new non-equilibrium structure, as observed underelectric field [82]. Soft glassy nature of the amorphous blue phase due to the disordered disclination
network is also an interesting issue related to the defect-mediated rheology. Melting of the amorphous
blue phase due to shear can be explained by the proliferation of disclinations as in the lyotropic hexagonalphase [23].
Thanks to the development in visualization technology and microrheology method, better
understanding of the stress response due to meso-scale structures has become possible in recent
years [22,83–87]. Simultaneous measurements of the viscoelasticity and image acquisition will give
us a time resolved spatial information under shear [77–79,88]. These technological developments are
expected to contribute to the fundamental science of non-equilibrium soft matter in the future.
Acknowledgments
We thank Y. Ishii for useful discussion, and T. Takahashi, H. Orihara, M. Imai, and Anton Paar Co.
Ltd. for the use of their rheometer and the microscopy devices. This work was mainly supported by
Grant-in-Aid for Scientific Research (B) grant No. 25287107 from MEXT of Japan. Shigeyuki Komura
would like to further acknowledge support from the Grant-in-Aid for Scientific Research on Innovative
Areas “Fluctuation & Structure” (grant No. 25103010), Grant-in-Aid for Scientific Research (C) grantNo. 24540439 from the MEXT of Japan, and the JSPS Core-to-Core Program “International research
network for non-equilibrium dynamics of soft matter”.
Author Contributions
Shuji Fujii performed the experiments. Shigeyuki Komura gave the concept on the structural rheology.
Chun-Yi David Lu gave pertinent comments on the data analysis and interpretations. Shuji Fujii
and Shigeyuki Komura wrote the manuscript. All the authors discussed the results and analysis andcommented on the manuscript.
Conflicts of Interest
The authors declare no conflict of interest.
References
1. Doi, M.; Edwards, S.F. The Theory of Polymer Dynamics; Clarendon Press: Oxford, UK, 1986.

Materials 2014, 75164

2. Cohen-Addad, S.; Pitois, O.; Höhler, R. Flow in foams and flowing foams. Annu. Rev.
Fluid Mech. 2013, 45, 241–267.
3. Lespiat, R.; Cohen-Addad, S.; Höhler, R. Jamming and flow of randomly close packed spherical
bubbles: An analogy with granular materials. Phys. Rev. Lett. 2011, 106, 148320,
doi:10.1103/PhysRevLett.106.148302.
4. Besseling, R.; Isa, L.; Weeks, E.R.; Poon, W.C.K. Quantitative imaging of colloidal flows.
Adv. Colloid Int. Sci. 2009, 146, 1–17.
5. Mason, T.G. New fundamental concepts in emulsion rheology. Curr. Opin. Colloid Int. Sci.
1999, 4, 231–238.
6. Sollich, P.; Lequeux, F.; Hebraud, P.; Cates, M.E. Rheology of soft glassy materials. Phys. Rev.
Lett. 1997, 78, 2020–2023.
7. Yamamoto, J.; Tanaka, H. Shear effects on layer undulation fluctuations of a hyper-swollen
lamellar phase. Phys. Rev. Lett. 1995, 74, 932–935.
8. Tamate, R.; Yamada, K.; Vinals, J.; Ohta, T. Structural rheology of microphase separated diblock
copolymers. J. Phys. Soc. Jpn. 2008, 77, 034802:1–034802:6.
9. Eskimergen, R.; Mortensen, K.; Vigild, M.E. Shear instability of a gyroid diblock copolymer.
Macromolecules 2005, 38, 1286–1291.
10. Fredrickson G.H.; Bates, F.S. Dynamics of block copolymers: Theory and experiments.
Annu. Rev. Mater. Sci. 1996, 26, 501–550.
11. Diat, O.; Roux, D. Effect of shear on dilute sponge phase. Langmuir 1995, 11, 1392–1395.
12. Porcar, L.; Hamilton, W.A.; Butler, P.D. Scaling of shear-induced transformations in membranephases. Phys. Rev. Lett. 2002, 89, 168301, doi:10.1103/PhysRevLett.89.168301.
13. Cui, C.T.; Cummings, H.D.; Cochran, H.D. Molecular simulation of the transition from liquid
like to solid like behavior in complex fluids confined to nanoscale gaps. J. Chem. Phys. 2001,
114, 7189, doi:10.1063/1.1359736.
14. Jabbarzadeh, A.; Harrowell, P.; Tanner, R.I. Crystal bridges, tetra tic order, and elusive equilibria:
The role of structure in lubrication films. J. Phys. Chem. B 2007, 111, 11354–11365.
15. Jabbarzadeh, A.; Tanner R.I. Thin lubricant films confined between crystalline surfaces: Goldversus mica. Tribol. Intern. 2011, 44, 711–719.
16. Fujii, S.; Ishii, Y.; Komura, S.; Lu, C.-Y.D. Smectic rheology close to the smectic-nematictransition. EPL 2010, 90, 64001, doi:10.1209/0295-5075/90/64001.
17. Fujii, S.; Komura, S.; Ishii, Y.; Lu, C.-Y.D. Elasticity of smectic liquid crystals with focal conic
domains. J. Phys.: Condens. Matter 2011, 23, 235105, doi:10.1088/0953-8984/23/23/235105.
18. Fujii, S.; Komura, S.; Lu, C.-Y.D. Structural rheology of focal conic domains: Stress-quench
experiment. Soft Matter 2014, 10, 5289–5295.
19. Fujii, S. Structural rheology of smectic liquid crystalline phase. J. Soc. Rheol. Jpn. 2012, 40,
229–237. (In Japanese)
20. Jones, J.L.; McLeish, T.C.B. Rheological response of surfactant cubic phases. Langmuir 1995,
11, 785–792.
21. Radiman, S.; Toprakcioglu, C.; McLeish, T.C.B. Rheological study of ternary cubic phases.
Langmuir 1994, 10, 61–67.

Materials 2014, 75165

22. Mohammad, M.A.; Mezzenga, R. Particle tracking microrheology of lyotropic liquid crystals.
Langmuir 2011, 27, 6171–6178.
23. Ramos, L.; Molino, F. Shear melting of a hexagonal columnar crystal by proliferation of
dislocations. Phys. Rev. Lett. 2004, 92, 018301, doi:10.1103/PhysRevLett.92.018301.
24. Colby, R.H.; Ober, C.K.; Gillmor, J.R.; Connelly, R.W.; Duong, T.; Galli, G.; Laus, M. Smectic
rheology. Rheol. Acta 1997, 36, 498–504.
25. Colby, R.H.; Nentwich, L.M.; Clingman, S.R.; Ober, C.K. Defect-mediated creep of structured
materials. Europhys. Lett. 2001, 54, 269, doi:10.1209/epl/i2001-00305-x.
26. Panizza, P.; Archambault, P.; Roux, R. Effects of shear on the smectic-A phase of thermotropic
liquid crystals. J. Phys. II France 1995, 5, 303–311.
27. Safinya, C.R.; Sirota, E.B.; Plano, R.J. Nematic to smectic-A phase transition under shear flow:
A nonequilibrium synchrotron x-ray study. Phys. Rev. Lett. 1991, 66, 1986–1989.
28. Larson, R.G.; Wineyl, K.I.; Patel, S.S.; Watanabe, H.; Bruinsma, R. The rheology of layered
liquids: Lamellar block copolymers and smectic liquid crystals. Rheol. Acta 1993, 32, 245–253.
29. Dhez, O.; Nallet, F.; Diat, O. Influence of screw dislocations on the orientation of a sheared
lamellar phase. Europhsy. Lett. 2001, 55, 821–826.
30. Diat, O.; Roux, D.; Nallet, F. Effect of shear on a lyotropic lamellar phase. J. Phys. II France
1993, 3, 1427–1452.
31. Nettesheim, F.; Zipfel, J.; Olsson, U.; Renth, F.; Lindner, P.; Richtering, W. Pathway of theshear-induced transition between planar lamellae and multilamellar vesicles as studied bytime-resolved scattering techniques. Langmuir 2003, 19, 3603–3618.
32. Fujii, S.; Koschoreck, S.; Lindner, P.; Richtering, W. Influence of a triblock copolymer onphase behavior and shear-induced topologies of a surfactant lamellar phase. Langmuir 2009,
25, 5476–5483.
33. Fujii, S.; Mitsumasu, D.; Isono, Y.; Richtering, W. Shear-induced onion formation ofpolymer-grafted lamellar phase. Soft Matter 2012, 8, 5381–5390.
34. Kosaka, Y.; Ito, M.; Kawabata, Y.; Kato, T. Lamellar-to-onion transition with increasing
temperature under shear flow in a nonionic surfactant/water system. Langmuir 2010, 26,3835–3842.
35. Horn, R.G.; Kleman, M. Observation on shear-induced textures and rheology of a smectic-A
phase. Ann. Phys. 1978, 3, 229–234.
36. Meyer, C.; Asnacios, S.; Bourgaux, C.; Kleman, M. Rheology of lyotropic and thermotropic
lamellar phases. Rheol. Acta 2000, 39, 223–233.
37. Meyer, C.; Asnacios, S.; Kleman, M. Universal properties of lamellar systems under weak shear.
Eur. Phys. J. E 2001, 6, 245–254.
38. Lu, C.-Y.D.; Chen, P.; Ishii, Y.; Komura, S.; Kato, T. Non-linear rheology of lamellar liquid
crystals. Eur. Phys. J. E 2008, 25, 91–102.
39. Basappa, G.; Suneel; Kumaran, V.; Nott, P.R.; Ramaswamy, S.; Naik, V.M.; Rout, D. Structure
and rheology of the defect-gel states of pure and particle-dispersed lyotropic lamellar phases.
Eur. Phys. J. B 1999, 12, 269–276.

Materials 2014, 7

5166

40. Ramos, L.; Zapotocky, M.; Lubensky, T.C.; Weitz, D.A. Rheology of defect networks incholesteric liquid crystals. Phys. Rev. E 2002, 66, 031711, doi:10.1103/PhysRevE.66.031711.
41. Medronho, B.; Migule, M.G.; Olsson, U. Viscoelasticity of a nonionic lamellar phase. Langmuir
2007, 23, 5270–5274.
42. Medronho, B.; Rodrigues, M.; Migule, M.G.; Olsson, U.; Schmidt, C. Shear-induced defectformation in a nonionic lamellar phase. Langmuir 2010, 26, 1477–1481.
43. Kleman, M.; Lavrentovich, O.D. Soft Matter Physics: An Introduction; Springer:
Berlin/Heidelberg, Germany, 2002.
44. Williams, C.E.; Kleman, M. Dislocations, grain boundaries and focal conics in smectic-A. J. Phys.
1975, 36, C1:315–C1:320.
45. Boltenhagen, P.; Kleman, M.; Lavrentovich, O.D. Freeze-fracture observations in the Lα phase ofa swollen surfactant in the vicinity of the L3 and the L1 phase transitions J. Phys. II France 1994,
4, 1439–1448.
46. Boltenhagen, P.; Lavrentovich, O.D.; Kleman, M. Oily streaks and focal conic domains in Lα
lyotropic liquid crystals. J. Phys. II France 1991, 1, 1233–1252.
47. Kleman, M.; Meyer, C.; Nastishin, Y.A. Imperfections in focal conic domains: The role ofdislocations. Philos. Mag. 2006 86, 4439–4458.
48. Meyer, C.; Nastishin, Y.; Kleman, M. Kinked focal conic domains in a SmA. Mol. Cryst. Liq.
Cryst. 2007 477, 43/[537]–53/[547].
49. Holyst, R. Dislocations in lamellar and liquid crystal films: Equilibrium location, edge profiles,
and phase transitions. Phys. Rev. Lett. 1994, 26, 4097–4100.
50. Helfrich, W. Defect model of the smectic-A-nematic phase transition. J. Phys. 1978, 39,
1199–1208.
51. Benzekri, M.; Marcerou, J.P.; Nguyen, H.T.; Rouillon, J.C. Critical behavior of the layer
compressional elastic constant B at the smectic-A-nematic phase transition. Phys. Rev. B 1990,
41, 9032–9037.
52. Benzekri, M.; Claverie, T.; Marcerou, J.P.; Rouillon, J.C. Nonvanishing of the layer compressionalelastic constant at the smectic-A-to-nematic phase transition: A consequence of Landau-Peierls
instability? Phys. Rev. Lett. 1992, 68, 2480–2483.
53. Nelson, D.R.; Torner, J. Bond-orientational order, dislocation loops, and melting of solids andsmectic-A liquid crystals. Phys. Rev. B 1981, 24, 363–387.
54. Moreau, P.; Navailles, L.; Giermanska-Kahn, J.; Mondain-Monval, O.; Nallet, F.; Roux, D.
Dislocation-loop-mediated smectic melting. EPL 2006, 73, 49–54.
55. De Gennes, P.G.; Prost, J. The Physics of Liquid Crystals; Clarendon Press: London, UK, 1993.
56. Davidov, D.; Safinya, C.R.; Kaplan, M.; Dana, S.S.; Schaetzing, R.; Birgeneau, R.J.; Litster, J.D.
High-resolution x-ray and light-scattering study of critical behavior associated with thenematic-smectic-A transition in 4-cyano-4’-octylbiphenyl Phys. Rev. B 1979, 19, 1657–1663.
57. Herschel, W.H.; Bulkley, B. Measurement of consistency as applied to rubber-benzene solution.
Proc. Am. Assoc. Test Mater. 1926, 26, 621–633.
58. Moller, P.C.F.; Mewis, J.; Bonn, D. Yield stress and thixotropy: On the difficulty of measuring
yield stresses in practice. Soft Matter 2006, 2, 274–283.

Materials 2014, 75167

59. Moller, P.; Fall, A.; Chikkadi, V.; Derks, D.; Bonn, D. An attempt to categorize yield stress fluid
behavior. Philos. Trans. R. Soc. A 2009, 367, 5139–5155.
60. Zywocinski, A.; Picano, F.; Oswald, P.; Geminard, J.C. Edge dislocation in a vertical smectic-A
film: Line tension versus temperature and film thickness near the nematic phase. Phys. Rev. E
2000, 62, 8133–8140.
61. Negita, K.; Kaneko, H. Rheodielectric study on shear-induced structural change in thesmectic-A phase of 4-n-octyl-4’-cyanobiphenyl (8CB). Phys. Rev. E 2009, 80, 011705,
doi:10.1103/PhysRevE.80.011705.
62. Bruinsma, R.; Rabin, Y. Shear-flow enhancement and suppression of fluctuations in smectic liquid
crystals. Phys. Rev. Lett. 1992, 45, 994–1008.
63. Goulian, M.; Milner, S.T. Shear alignment and instability of smectic phases. Phys. Rev. Lett.
1995, 74, 1775–1778.
64. Ramaswamy, S. Shear-induced collapse of the dilute lamellar phase. Phys. Rev. Lett. 1992, 69,
112–115.
65. Porcar, L.; Warr, G.G.; Hamilton, W.A.; Butler, P.D. Shear-induced collapse in a lyotropic
lamellar phase. Phys. Rev. Lett. 2005, 95, 078302, doi:10.1103/PhysRevLett.95.078302.
66. Berghausen, J.; Zipfel, J.; Lindner, P.; Richtering, W. Shear-induced orientations in a lyotropic
defective lamellar phase. Europhys. Lett. 1998, 43, 683–689.
67. van der Linden, E.; Droege, J.H.M. Deformability of lamellar droplets. Physica A 1993, 193,
439–447.
68. van der Linden, E.; Hogervorst, W.T.; Lekkerkerker, H.N.W. Relation between the size of lamellar
droplets in onion phases and their effective surface tension. Langmuir 1996, 12, 3127–3130.
69. Princen, H.M.; Kiss, A.D. Rheology of foams and highly concentrated emulsions: III. Static shear
modulus. J. Colloid Interface Sci. 1986, 112, 427–437.
70. Princen, H.M.; Kiss, A.D. Rheology of foams and highly concentrated emulsions: IV. An
experimental study of the shear viscosity and yield stress of concentrated emulsions. J. Colloid
Interface Sci. 1989, 128, 176–187.
71. Panizza, P.; Roux, D.; Vuillaume, V.; Lu, C.-Y.D.; Cates, M. Viscoelasticity of the onion phase.
Langmuir 1996, 12, 248–252.
72. Leng, J.; Nallet, F.; Roux, D. Anomalous elasticity of an ordered lamellar liquid foam. Eur. Phys.
J. E 2001, 4, 337–342.
73. Chatterjee, S.; Anna, S.L. Formation and ordering of topological defect arrays produced bydilatational strain and shear flow in smectic-A liquid crystals. Phys. Rev. E 2012, 85, 011701,
doi:10.1103/PhysRevE.85.011701.
74. Chatterjee, S.; Anna, S.L. Interaction of toroidal focal conic defects with shear flow. Soft Matter
2012, 8, 2698–2705.
75. Oswald, P.; Kleman, M. Experimental evidence for helical instability of screw dislocation lines ina smectic-A phase. J. Phys. Lett. (France) 1984, 45, L319–L328.
76. Oswald, P.; Kleman, M. Lubrication theory of smectic-A phases. J. Phys. Lett. (France) 1982,
43, L411–L415.

Materials 2014, 7

5168

77. Aida, K.; Na, Y.H.; Nagaya, T.; Orihara, H. Droplet coalescence process under electric fields in
an immiscible polymer blend. Phys. Rev. E 2010, 82, 031805, doi:10.1103/PhysRevE.82.031805.
78. Aida, K.; Na, Y.H.; Nagaya, T.; Orihara, H. Storage shear modulus of columnar structure formed
in an immiscible polymer blend under electric fields. Phys. Rev. E 2009, 80, 041807:1–041807:5.
79. Orihara, H.; Takikawa, Y. Brownian motion in shear flow: Direct observation of anomalous
diffusion. Phys. Rev. E 2011, 84, 061120:1–061120:5.
80. Henrich, O.; Stratford, K.; Cates, M.; Marenduzzo, D. Structure of blue phase III of cholestericliquid crystals. Phys. Rev. Lett. 2011, 106, 107801:1–107801:4.
81. Henrich, O.; Stratford, K.; Coveney, P.V.; Cates, M.; Marenduzzo, D. Rheology of cubic bluephases. Soft Matter 2013, 9, 10243–10256.
82. Kitzerow, H.-S.; Crooker, P.P.; Heppke, G. Line shapes of filed-induced blue-phase III selective
reflections. Phys. Rev. Lett. 1991, 67, 2151–2154.
83. Mason, T.G.; Weitz, D.A. Optical measurement of frequency-dependent linear viscoelastic moduli
of complex fluids. Phys. Rev. Lett. 1995, 74, 1250–1253.
84. Crocker, J.C.; Valentine, M.T.; Weeks, E.; Gisler, T.; Kaplan, P.D.; Yodh, A.G.; Weitz, D.A.
Two-point microrheology of inhomogeneous soft materials. Phys. Rev. Lett. 2000, 85, 888–891.
85. Sonn-Segev, A.; Groswasser, A.B.; Diamant, H.; Roichman, Y. Response of a compilex fluid atintermediate distances. Phys. Rev. Lett. 2014, 112, 088301:1–088301:5.
86. Kimura, Y.; Mizuno, D. Microrheology of a swollen lyotropic lamellar phase. Mol. Cryst.
Liq. Cryst. 2007, 478, 759–769.
87. Yamamoto, N.; Ichikawa, M.; Kimura, Y. Local mechanical properties of a hyperswollen lyotropic
lamellar phase. Phys. Rev. E 2010, 82, 021506, doi:10.1103/PhysRevE.82.021506.
88. Dutta, S.K.; Mbi, A.; Arevalo, R.C.; Blair, D.L. Development of a confocal rheometer for softand biological materials. Rev. Sci. Instrum. 2013, 84, 063702:1–063702:7.
c 2014 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article
distributed under the terms and conditions of the Creative Commons Attribution license(http://creativecommons.org/licenses/by/3.0/).