June 2010
EPL, 90 (2010) 64001
doi: 10.1209/0295-5075/90/64001

www.epljournal.org

Smectic rheology close to the smectic-nematic transition
S. Fujii1(a) , Y. Ishii2 , S. Komura3 and C.-Y. D. Lu4
1

Department of Chemistry, Nagaoka University of Technology - Niigata 940-2188, Japan
Department of Physics and Applied Physics, Waseda University - Tokyo 169-8555, Japan3
Department of Chemistry, Graduate School of Science and Engineering, Tokyo Metropolitan University
Tokyo 192-0397, Japan
4
Department of Chemistry, National Taiwan University - Taipei 106, Taiwan
2

received 10 February 2010; accepted in ﬁnal form 15 June 2010
published online 30 June 2010
47.57.Qk – Complex ﬂuids and colloidal systems: Rheological aspects
64.70.mj – Experimental studies of liquid crystal transitions
PACS 61.30.Jf – Defects in liquid crystals
PACS
PACS

Abstract – We study the rheological properties of a thermotropic liquid crystal, 8CB, in thesmectic phase close to the smectic-nematic (Sm-N) transition temperature. Three diﬀerent regimes
were identiﬁed in the ﬂow curves at diﬀerent temperatures: i) appearance of the yield stress at
low stresses, ii) power law behavior at intermediate shear stresses, and iii) Newtonian at higher
shear stresses. The vanishing of the yield stress at the Sm-N transition temperature is correlatedwith a rapid growth of focal conic domains. The constructed dynamic phase diagram exhibits thetwo diﬀerent smectic phases together with the ﬂow-induced Sm-N transition.
c EPLA, 2010
Copyright Introduction. – The rheology of smectic liquid crystals is a long-standing puzzle [1–3]. Although the theory
predicts a liquid response when the smectic layers arealigned parallel to the shear plane, some experiments
showed that the system behaves as a solid until a critical shear stress is exceeded [4–6]. This means that eventhe oriented smectic liquid crystal behaves as a yieldstress ﬂuid [7] at very low stress. Such a peculiar behavior can be attributed to the presence of defects such as
focal conic domains in the smectic layers, because they
will act to hinder layer sliding in the low stress limit.
Indeed it was seen that the focal conic domain density
strongly correlates with the viscosity [4]. At stress levels
slightly above the yield stress, there is a yielding regime
and the smectic phase starts to ﬂow [5,6]. In general,the yield stress of a stationary sample ﬂuctuates considerably and can be highly dependent on its ﬂow history
such as a sample loading process. Meyer et al. proposeda theory of smectic rheology which is explicitly based on
the screw defect line motion [8,9]. By using the Orowan
relation [10], they obtained a power law behavior of theconstitutive relation between the shear rate γ̇ and theshear stress σ such that γ̇ ∼ σ m with a universal exponent m = 5/3. This shear thinning (m > 1) behavior was
conﬁrmed not only for a thermotropic smectic liquid(a) E-mail: sfujii@mst.nagaokaut.ac.jp

crystal, but also in the lamellar phases of a copolymer solution and a lyotropic system [8,9]. Later a slightly diﬀerent
exponent m = 3/2 was predicted by some of the present
authors who considered the motion of dislocation loops
under shear ﬂow which causes the continuous productionof dislocations [11].
On the other hand, the proliferation of dislocations
takes place close to the smectic-nematic (Sm-N) transition
point even in the absence of any applied external ﬁeld, i.e,in thermal equilibrium. This is known as the dislocationloop-mediated smectic melting or the dislocation unbinding transition proposed by Helfrich in 1978 [12] and further
developed by other scientists [13]. For smectic systems, theunbinding of the dislocation loop is described as the divergence of the defect size. Recently, the appearance of dislocation loops was directly observed using freeze-fracturetransmission electron microscopy in a lyotropic systemin the vicinity of the smectic-cholesteric (chiral nematic)transition [14]. Although the concept of topological melting is quite important in many condensed-matter systems,
its experimental evidence is very limited. Especially, it istotally unknown how the unbinding of dislocation loops
aﬀects the average size of focal conic domains which play
an important role in the smectic rheology. The aim of
the present letter is to investigate the rheological properties of a thermotropic liquid crystal (8CB) in the smectic
phase close to the Sm-N transition temperature. This is an64001-p1

S. Fujii et al.

Experimental. – We used the thermotropic liquidcrystal 4-n-octyl-4 -cyanobiphenyl (8CB) as a smectic
liquid crystal in this study. The phase behavior and somephysical properties of 8CB have been studied in the literatures [15,17]. 8CB was obtained from SYNTHON Chemicals GmbH & Co., Germany and was used without further
puriﬁcation. Rheological measurements were performed
using the stress-controlled rheometer, Anton Paar MCR300, equipped with a cone-plate shear cell which has adiameter of 50 mm and a cone angle of 1 degree. In contrast
to ref. [4], no surface treatment has been done in our
experiment. The lack of the homeotropic alignment ofsmectic layers may easily induce the nucleation of focal
parabolae under shear ﬂow. Temperature was controlled
within 0.02 K by a Peltier device attached to the rheometer. We determined the smectic-A–to–nematic transition
temperature TSN by the dynamic viscoelastic measurement. From the condition that the storage modulus disappears, we obtained TSN = 33.4 ◦ C. This value is in good
agreement with that reported previously [15]. In the rheological measurement, the shear stress was imposed andthe shear rate was measured. The subsequent stress sweep
proceeded from high to low stresses. Thus the samples

4

10

3

10

2

10

II

I

3 101

10

0

10

-1

2

10

-1

γ [s ]

interesting situation because the proliferation of dislocation loops is induced by both equilibrium thermal
ﬂuctuations and non-equilibrium shear ﬂow. One cannaturally expect that the eﬀects of temperature and ﬂow
are strongly correlated so that a very unique rheologicalresponse could be observed as the transition is approachedfrom below. Our rheology measurement reveals that the
Sm-N transition temperature is lowered in the presence ofan applied shear stress. We also address the issue of theuniversal shear thinning exponent m in the smectic phasefar from the transition point as well as the ﬂow-inducedorientation transition. To our knowledge, the present workis the ﬁrst systematic study on the smectic rheology overwide shear stress and temperature ranges.
Before explaining our results, we brieﬂy refer to someof the additional related works. As the N-Sm transition
is approached from above in the nematic phase undershear ﬂow, Saﬁnya et al. showed that one of the viscosities
diverges due to the critical slowing-down of the smectic
order-parameter ﬂuctuations [15,16]. When the temperature of the smectic phase is not very close to thetransition point, Panizza et al. performed X-ray scatteringstudies under shear ﬂow [17]. They found that a discontinuous dynamic transition separates between the two smectic orientations exhibiting diﬀerent rheological behaviors;
the system is shear thinning (m = 2) at low shear rates,
while it becomes Newtonian (m = 1) at high shear rates.
More recently, the frequency-dependent shear modulus of
the smectic liquid crystal in a colloidal aerosil gel was
analyzed [18] using the concept of soft glassy rheology [19].
For a lyotropic lamellar phase, on the other hand, it wasalso shown that the proliferation of dislocation strongly
aﬀects the rheological and orientational properties [20,21].

.

10

-2

10 -1
10

σt

0
0

10

10

1

o

34 C
33.5
33.4
33.3
33.2
33.1
33
32.9
32.8
32.5
32
31
29
27
25

2

10

1

10

0

10

-1

10

-2

10 -2
10

-1

10

0

10

1

10

σ [Pa]

2

10

3

10

Fig. 1: (Colour on-line) Steady-state shear rate γ̇ as a functionof 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-Bulkely model,
eq. (1); Regime I is ﬁtted by the power law behavior, eq. (2);and Regime II corresponds to the Newtonian behavior.
σt indicates a stress where the transition from Regime I to
Regime II takes place.

were always pre-sheared before the measurement at every
shear stress such that the defect density is well deﬁned.
Each shear stress was applied for 1000 s because the steadystate was achieved within 600 s. The steady-state shear
rate was averaged over 600–1000 s at every ﬁxed shear
stress. We also measured the temperature dependence ofthe shear viscosity under diﬀerent applied shear stresses.
Here the temperature was swept with the heating rate of
0.05 K/min from low to high values.
Microscope observation under shear ﬂow was performedby using a Linkam CSS 450 shear cell which has aplate-plate shear geometry attached onto a Olympus
microscope, BX-50, between crossed polarizers. The gap
size between the two plates was ﬁxed to 150 µm.
Results and discussion. –Flow curves. Figure 1 shows the applied shear stress
σ-dependence of the steady-state shear rate γ̇ measured
at diﬀerent temperatures ranging from 25 to 34 ◦ C (thelatter being slightly above TSN = 33.4 ◦ C). We ﬁrst notethat the slope of these ﬂow curves was larger than unity
for all the temperatures in the smectic phase, indicatinga shear thinning behavior. However, these data cannot be
expressed by a single power law behavior with a universal
exponent m. Especially, the ﬂow curves in the smectic
phase exhibited non-zero shear stress, or the yield stress
σy , in the limit of γ̇ → 0. As we shall describe below

64001-p2

Smectic rheology close to the Sm-N transition

σ = C γ̇ 1/m ,(2)

where C is a pre-factor. This equation can be considered as
the special case of eq. (1) be setting σy = 0 and n = 1/m.
In Regime II, the smectic phase becomes Newtonian with
m = 1. All the other ﬂow curves could be also well ﬁtted by
the HB model (Regime 0) and the power law separately(Regime I), although Regime II was not observed when
the temperature is low.
The temperature dependence of σy , A, n (Regime 0),
C, m (Regime I) obtained by the best ﬁt to eqs. (1) and(2), respectively, are summarized in ﬁg. 2. We see that
σy and A showed a remarkable temperature dependence
especially in the vicinity of the Sm-N transition, while theexponent n depends only weakly on temperature. Noticethat both σy and A vanish at T = TSN , and the values ofn are always less than unity. On the other hand, both C
and m were almost constant up to 32 ◦ C, but increased
remarkably in the vicinity of the Sm-N transition. The
increase in m implies that the degree of shear thinning
is more enhanced close to the transition. The constantvalue of m ≈ 1.7 at low temperatures is almost consistent
with the experimental result m = 1.67 and the theoretical

σy [Pa]

1.0
0.5
0.0

A

4
2
0

n

1.0
0.5
0.0
25 26 27 28 29 30 31 32 33 34 35

C

4
2
0
3

m

in more detail, the whole behavior can be describedby a combination of three diﬀerent regimes. When thetemperature is increased toward TSN , the curve shifted tolower shear stresses. Above TSN in the nematic phase, only
the Newtonian behavior with a single slope m = 1 could
be found. An interesting behavior observed at higher
temperatures within the smectic phase is the rheologicaltransition from the shear thinning to the Newtonian
behavior at higher shear stresses. The threshold shear
stress for this rheological transition, denoted by σt ,shifted to lower stresses as the temperature was increased.
A similar dynamic transition was reported in refs. [15,17]
in which the authors attributed it to the shear-inducedorientational transition of the smectic layers [22] fromleek to perpendicular alignment. We also note that thesteady shear data for 8CB at 26 ◦ C in refs. [5,6] are inagreement with those in ﬁg. 1.
Following the argument in ref. [6], the ﬂow curves inthe smectic phase were divided into three regimes; shear
thinning with the yield stress at low stresses (Regime 0),
shear thinning at intermediate shear stresses (Regime I),
and Newtonian at higher shear stresses (Regime II).
Note that σt separates between the Regimes I and II.
A representative ﬂow curve for T = 33.1 ◦ C is shown in
the inset of ﬁg. 1. In order to quantify the yield stressin Regime 0, the data were ﬁtted with the HerschelBulkely (HB) model describing a pseudo viscoplastic
behavior(1)σ = σy + Aγ̇ n ,where σy is the yield stress, A and n are adjustableparameters [7]. The best ﬁt to the HB model is shownin the inset. In the intermediate Regime I, the data wereﬁtted with a power law behavior as described in the
introduction [6,8,9];2
1
25 26 27 28 29 30 31 32 33 34 35o

T [ C]

Fig. 2: (Colour on-line) The yield stress σy , the pre-factor
A, and the exponent n in eq. (1), the pre-factor C and theshear thinning exponent m in eq. (2) as a function of thetemperature. These values are obtained by the best ﬁt in
ﬁg. 1 for each regime. Vertical dotted line indicates the Sm-N
transition temperature; TSN = 33.4 ◦ C.

prediction m = 5/3 in refs. [8,9]. A slightly larger exponent
m = 2 was reported for 8CB at 31 ◦ C [17]. It should be
noted, however, that the determination of m includes somearbitrariness depending on how we identify the range ofeach regime. The constant viscosity measured in Regime II
was about 0.045 Pa s. This value is in reasonable agreementwith the previously obtained value 0.036 Pa s [17]. We also
note that it is close to the viscosity value measured in thenematic phase.
In ref. [4], Horn and Kleman reported that 8CB behaves
as a Bingham ﬂuid at 25 ◦ C, i.e., A = ηapp and n = 1 (ηappis the apparent viscosity). Such a diﬀerent exponent can be
attributed to the imposed homeotropic alignment of thesmectic sample in their measurement. In our experiment,
the imperfect alignment of the homeotropic anchoring may

64001-p3

S. Fujii et al.

50

a

b

c

d

e

f

L [µm]

40
30
20
10
0
25 26 27 28 29 30 31 32 33 34 35o

T [ C]
Fig. 3: The size of focal domains L as a function of thetemperature. We have used eq. (3), and the curvature elastic
constant K is taken to be constant over the temperature range
performed in this study. The vertical dotted line indicates
T = TSN .

enhance the nucleation of focal conic domains under shear
ﬂow. It should be stressed, however, that the obtained ﬂow
curves in ﬁg. 1 are completely reproducible even without
any surface treatment.
We note that the Newtonian behavior of Regime II inﬁg. 1 has nothing to do with the Newtonian behavior in
Regime II discussed in ref. [6]. In the latter case, viscosity
is independent of shear rate only because molecularmotion keeps the molecular conﬁguration undisturbed by
the ﬂow if γ̇R2 /D < 1, where R is the molecular size and D
is the diﬀusion coeﬃcient. In the present case, the smectic
becomes Newtonian when the layers align perpendicular tothe shear ﬂow, as we will discuss later. Hence the constantviscosity 0.045 Pa s obtained in Regime II corresponds toη2 , i.e., the shear viscosity in the plane of the layers [2].
Focal conic domain size. For stresses lower than σy ,our view is that smectic behaves as an elastic solid witha modulus determined by the defect size. Above the yieldstress, on the other hand, the motion of defects would
dominate the rheological properties. According to Hornand Kleman [2,4], the yield stress was related to the focal
conic domain size L by a dimensional analysis as follows:
σy ≈

K
.
L2

(3)

Here, K is the curvature elastic modulus [23]. Since K
is known to be almost constant around the transition
temperature with the value K = (5.2 ± 0.3) × 10−12 N [24],we are able to extract the temperature dependence of L

Fig. 4: Top: polarized microscopic images at a shear rate of γ̇ =1 s−1 for (a) T = 25 ◦ C, (b) 29 ◦ C and (c) 31 ◦ C, respectively.
Bottom: polarized microscopic images at 33 ◦ C under a shearof (d) γ̇ = 0.1 s−1 , (e) 1 s−1 and (f) 10 s−1 , respectively. Flow
is applied along the longitudinal direction. The scale barcorresponds to 100 µm.

as shown in ﬁg. 3. Here we see a remarkable growth of
the focal conic domain size L as TSN is approached frombelow.
Figure 4 shows the polarized microscopic images
observed at diﬀerent temperatures and diﬀerent applied
shear rates. Along the ﬂow direction, oily streaks areformed due to the stacking of focal conic domains.
A similar texture under ﬂow was shown in ﬁg. 5 of ref. [9].
When comparing between ﬁgs. 4(a)–(c) and (e) at ﬁxed
shear rate γ̇ = 1 s−1 , we observe that the width of the oily
streak becomes larger as the temperature is increased
towards TSN . The size of the focal conic domain is almostin accord with the estimated size L in ﬁg. 3. Hence it isreasonable to consider that the vanishing of σy at TSN in
ﬁg. 2 is closely related to the rapid growth of L as shownin ﬁgs. 3 and 4. The present result is an indication of thedislocation unbinding behavior, because the growth of
the focal conic domains close to the transition point may
be caused by the increment of the dislocation loops [14].
Although we do not have any direct evidence for theincreased dislocation loop density, more defects shouldsoften the system so that the large focal conic domains are
not strongly penalized by their elastic energy. Therefore
the rapid growth of the focal conic domain size close to
the transition temperature may reﬂect the proliferationof dislocation loops as predicted by the theory [12,23].
When the shear rate is increased at constant temperature as in ﬁgs. 4(d)–(f), on the other hand, we clearly seethat the width of oily streak decreases systematically. Thistendency is also consistent with the prediction that thesize of focal domains L is a decreasing function of γ̇
in Regime I, or more precisely, a power law behavior
L ∼ γ̇ −1/5 as discussed in refs. [8,9]. Although it is diﬃcult
to conﬁrm this relation quantitatively from the microscopeobservation, we ﬁnd that the defect density increases
for higher shear rates. This phenomena reﬂects the64001-p4

Smectic rheology close to the Sm-N transition2(a=0.6)(a=0.4)(a=0.2)
(a=0)

100
80

σ [Pa]log η + a

1

10 Pa
30
50
100

0

60

SmAI

SmAII

N

32

34

40
σt

-1
20

-2
30

T1
31

T2
32

o

33

34

0
30

35

T [ C]

T1
T2
31

o

33

35

T [ C]

Fig. 5: (Colour on-line) Shear viscosity η as a function of thetemperature obtained at diﬀerent applied shear stress σ = 10,30, 50, 100 Pa. Here log η is shifted by a constant a in order tohave a better visibility. Vertical dotted line indicates T = TSN .

proliferation of dislocation loops due to the nonequilibrium shear ﬂow. In Regime II, we could not observe
oily streaks at high shear rate where the Newtonian
behavior appeared.
Dynamic phase diagram.
In addition to the existing dislocations in thermal equilibrium, those induced byshear ﬂow might also aﬀect the rheological propertiesof the smectic phase. In order to construct a dynamic
phase diagram, we have measured the temperature dependence of the shear viscosity η under diﬀerent applied shear
stresses 10, 30, 50 and 100 Pa. The result is shown inﬁg. 5 in which η showed a rather complicated temperature dependence. In the low temperatures, the viscosity
decreased by increasing the temperature, followed by aplateau region starting at T1 . A further increase in thetemperature caused a peak at T2 slightly below TSN . Interestingly, both T1 and T2 shifted to lower temperatures as
the applied shear stress is increased.
In ﬁg. 6, the two characteristic temperatures T1 andT2 are plotted against the applied shear stress σ together
with the data of σt at which the transition from Regime I
to Regime II takes place (as determined in ﬁg. 1). In
this plot, we see that T1 and T2 varies linearly with σ,and so does σt . More importantly, the data of T1 and σt
coincide with each other. The extrapolations of T1 and T2
to zero shear stress give a temperature which is in accord
with TSN at the quiescent state. In the lower-temperature
region, T < T1 , where the viscosity exhibited a remarkabletemperature dependence, the smectic is shear thinning(Regime I) and is denoted as SmAI . In the intermediatetemperature region, T1 < T < T2 , where the viscosity isFig. 6: (Colour on-line) Dynamic phase diagram of 8CB
under shear plotted against the temperature and the applied
shear stress σ. σt is the stress above which the smectic
becomes Newtonian in ﬁg. 1, while T1 and T2 are characteristic
temperatures identiﬁed in ﬁg. 5. The vertical dotted line
indicates T = TSN . The lines of σt and T1 coincide with eachother. SmAI and SmAII denote the smectic phases in Regime
I and II, respectively, while N indicates the nematic phase.

almost constant, the smectic is Newtonian (Regime II) andis denoted as SmAII . These regions are marked in ﬁg. 6.
This rheological transition from the shear thinning(SmAI ) to the Newtonian behavior (SmAII ) can becompared with ﬁg. 11 in ref. [17]. In their case, theorientation diagram of the smectic phase consists of two
regimes: i) the shear thinning regime (m = 2) wherea leek structure is oriented along the ﬂow direction,ii) the Newtonian regime (m = 1) where the perpendicular orientation of layers is observed. Our result is almost inaccord with these behaviors, although the line separating
between these two regimes slightly deviates from ourdata of T1 or σt . As a consequence, the temperature atwhich the SmAI phase disappears in their case do notcoincide with TSN , and is about 1.6 ◦ C higher. Although
the system presented a hysteresis in ref. [17], we did notobserve any apparent discontinuous jump in the shear
rate when the transition from SmAI to SmAII takes
place. Our phase diagram is also consistent with theorientation transition discussed by Saﬁnya et al. [15].
In the SmAII regime, they observed the perpendicularorientation where the layer normal points along thevorticity direction. Since a cybotactic cluster encourages
the layer ﬂuctuation around the SmA-N transition [23], ascattering pattern, as observed by Panizza et al., suggeststhat the perpendicular orientation is attributed to thecybotactic cluster.

64001-p5

S. Fujii et al.
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∗∗∗
We acknowledge useful discussions with H. Orihara.
We thank T. Takahashi and M. Imai for allowing us to
use the rheometer MCR-300 and the Linkam shear cell
CSS-450, respectively. SF and SK acknowledge support
by KAKENHI (Grant-in-Aid for Scientiﬁc Research)on Priority Areas “Soft Matter Physics” and Grant
No. 21540420 from the Ministry of Education, Culture,
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