Eur. Phys. J. E 25, 91–101 (2008)
DOI 10.1140/epje/i2007-10267-3

THE EUROPEAN
PHYSICAL JOURNAL E

Non-linear rheology of lamellar liquid crystals
C.-Y.D. Lu1,a , P. Chen2 , Y. Ishii3 , S. Komura3,b , and T. Kato3
1

Department of Chemistry, National Taiwan University, Taipei 106, Taiwan
Department of Physics and Center for Complex Systems, National Central University, Jung-li 320, Taiwan
3
Department of Chemistry, Graduate School of Science and Engineering, Tokyo Metropolitan University, Tokyo 192-0397,
Japan

2

Received 1 August 2007
Published online: 29 February 2008 – c EDP Sciences / Società Italiana di Fisica / Springer-Verlag 2008
Abstract. We measure the non-linear relation between the shear stress and shear rate in the lyotropic
lamellar phase of C12 E5 /water system. The measured shear thinning exponent changes with the surfactant concentration. A simple rheology theory of a lamellar or smectic phase is proposed with a predictionγ̇ ∼ σ 3/2 , where γ̇ is the shear rate and σ is the shear stress. We consider that the shear flow passed throughthe defect structure causes the main dissipation. As the defect line density varies with the shear rate, theshear thinning arises. The defect density is estimated by the dynamic balance between the production andannihilation processes. The defect production is caused by the shear-induced layer undulation instability. The annihilation occurs through the shear-induced defect collision process. Further flow visualization
experiment shows that the defect texture correlates strongly with the shear thinning exponent.
PACS. 61.30.Jf Defects in liquid crystals – 83.60.Fg Shear rate dependent viscosity – 61.72.Lk Linear
defects: dislocations, disclinations

1 Introduction
The rheology of lamellar or smectic liquid crystals is along-standing puzzle [1]. A very thin free-standing layer
which is very well oriented shows a medium value in theplane shear viscosity (the η3 in the Harvard notation [2])of the order 0.03–0.07 Pa s for rod-like molecules such as
8CB [3]. Real flows in general situations, however, are totally different, where the samples become far more viscousthan the well-aligned ones. This is believed mainly due tothe texture. The presence of domains, together with grainboundaries and line defects between them, is responsible
for the high viscosity. Taking the finite grains as the basic
unit, Kawasaki and Onuki suggested that the layer relaxation within the lamellar domains may have an important
contribution to the linear viscoelasticity [4]. The detailed
grain boundary or defect structures do not enter into theirarguments, which indicates a high universality. Other theoretical suggestions examine the layer fluctuations within
the extremely large domain [5,6]. This fluctuation effect isestimated with a logarithmic dependence on the domain
size, and should be small for the typical sample which consists of small grains. The direct visualization by Basappa
et al. shows that the defect line density correlates strongly
with the viscosity [7]. Therefore the defects must play acentral role in the lamellar rheology.
abe-mail: cydlu@ntu.edu.twe-mail: komura@tmu.ac.jp

Recently Meyer et al. have proposed a theory for lamellar viscosity which is explicitly based on the defect line
motion [8–10]. They considered the screw defect line motion which produces the plastic deformation of the lamellar structure. The defect lines move between the sites of
local free-energy minima, which is responsible for the dissipation. Their argument arrives at a power law behavior
with the constitutive relation γ̇ ∼ σ m , with the universal
shear exponent m = 5/3 relating the shear rate γ̇ and theshear stress σ. (Here we use the notation in Refs. [9,10]
which follows the metallurgical usage. The conventional
shear thinning exponent n defined by σ ∼ γ̇ n is simply related by n = 1/m.) Three experimental systems, including a copolymer solution, a thermotropic smectic liquid
crystal (8CB), and a lyotropic system, all support thisexponent [10]. Other earlier and recent experiments on
smectics or lamellar phases reported that the exponent
m varies between 1 and 2 [11–16] (see Tab. 2 later). At
the moment, it is not clear whether the exponent is truly
universal and all these variations are due to experimental
errors, or it depends on the systems studied.
In this work, we first study experimentally the lyotropic system of a non-ionic surfactant C12 E5 and water. In Section 2, we discuss its non-linear rheological behavior as the temperature and concentration are varied.
We find that the C12 E5 /water binary mixture has a shear
exponent different from the systems studied by Meyer
et al. [9,10]. In Section 3 we propose a scaling argument

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2
0.6 Pa

0.15

0.4

0.09
1

γ [1/s]γ [1/s]

0.3

1

0.1

0

0

200

400

600

time [s]0.11
σ [Pa]

Fig. 1. (Color online) Time evolution of the measured shear
rate γ̇ under constant stresses σ = 0.09, 0.15, 0.3, 0.4, 0.6 Pa
during 600 s for a C12 E5 /water binary system. The surfactant
composition is φ = 35 wt% and the temperature is T = 66 ◦ C
at which the system is in the lamellar phase.

Fig. 2. Log-log plot of the steady-state shear rate γ̇ (averaged
over 450–600 s in Fig. 1) as a function of the shear stress σwhen φ = 35 wt% and T = 66 ◦ C. The slope of the straightline is m = 1.44 ± 0.08 which gives the exponent of the stressscaling behavior γ̇ ∼ σ m .

for the shear exponent which operates when the defect is
produced by the flow instability. We estimate the defectdensity by the dynamic balance between the production
and annihilation processes. Our theoretical prediction for
the exponent (m = 3/2) is in agreement with our experimental results. We reported in Section 4 a preliminary
flow visualization experiment which shows the correlation
between the shear exponent and the defect texture. Some
discussions and the comparison with the theory by Meyer
et al. are provided in Section 5.

We have checked that the system attains the stationary
state within 600 s by performing much longer measurements. For very small stress values such as σ = 0.09 Pa,
however, the small shear rate signal interferes with an artificial oscillatory noise generated by the rheometer, andit becomes more difficult to determine the stationary state
accurately. Nevertheless, we have compared the shear rate
data up to 600 s for all the stress values by assuming thatthe system has reached the steady state. The applied shear
stress was successively decreased from the higher to lower
values. The sample was kept at rest for 120 s between successive measurements with different stresses. The solvent
evaporation is insignificant within one sequence of the experimental time. Such a measurement has been repeatedseveral times and the obtained data were accumulated inorder to better estimate the scaling relation between theshear stress and shear rate. We have also tried to measure
the shear rate by successively increasing (rather than decreasing) the shear stress from the lower to higher values.
However, this was impossible without a preshearing treatment at the beginning. After preshearing the sample forseveral minutes, we have recovered almost the same result as presented above. Figure 2 is the log-log plot of thesteady-state shear rate (averaged between 450 and 600 s)
as a function of the applied stress for several sequencesof the measurement. As indicated by the solid fitted line,
the shear rate obeys a power law behavior γ̇ ∼ σ m , with
m = 1.44 ± 0.08.
Next we investigated the effect of temperature when
φ = 35 wt%. The measured shear rate is plotted for various temperatures in Figure 3. For each temperature, we
extract the exponent m which is summarized in Table 1.
For the studied temperature range, m is almost independent of the temperature within the experimental error.
Then the temperature effect can be incorporated in the
prefactor A(T ) such that γ̇ ∼ A(T )σ m , as suggested in

2 Rheology experiment
In this section, we first present the results of rheology
experiment on a lyotropic liquid crystal. The studied system is composed of a binary mixture of water and nonionic surfactant C12 E5 (Cn Em represents the chemical formula Cn H2n+1 (OCH2 CH2 )m -OH). We purchased C12 E5from Nikko Chemicals, Inc., and used without any further purification. The composition-temperature phase diagram of this system has been reported [17]. The rheological measurement was performed in the surfactant composition range φ = 30–45 wt% and the temperature range
T = 66–71 ◦ C. For this condition, the system is in the single lamellar phase. We used a rotational rheometer AR550
from TA Instruments, Ltd. All the measurements werecarried out under constant stress σ in the cone-and-plategeometry. Here the cone is made in stainless steel. The
diameter of the cone is 40 mm and the cone angle is 1.5◦ ,while the truncation is 47 µm. No anchoring treatment was
performed.
Typical time evolutions of the measured shear rate arepresented in Figure 1 for different applied shear stresses
when φ = 35 wt% and T = 66 ◦ C. Each stress was applied
for 600 s within which the sample reached the steady state.

C.-Y.D. Lu et al.: Non-linear rheology of lamellar liquid crystals

93

2.0

log A(T )γ [1/s]

1

344 K

1.5

343
341

0.1

1.0

339
0.1

1

0.00292

σ [Pa]

0.00294

1/ T [1/K]

Fig. 3. (Color online) Log-log plot of the steady-state shear
rate γ̇ as a function of the shear stress σ at different temperatures but with fixed composition φ = 35 wt%. Each slope of thestraight line is almost identical to that of T = 66 ◦ C (339 K)
within the experimental errors.

Fig. 4. Normal plot of log A(T ) as a function of 1/T when
φ = 35 wt%. The slope of the straight line gives the activationenergy W ≈ 65kB TR , with TR = 339 K (see Eq. (1) and thetext).

Table 1. The values of the stress scaling exponent m definedby γ̇ ∼ σ m for different temperatures when φ = 35 wt%.
m
1.44 ± 0.08
1.44 ± 0.07
1.42 ± 0.07
1.48 ± 0.09

1
γ [1/s]T (◦ C)
66
68
70
71

reference [9]. Here we assume that the macroscopic rheological behavior corresponds to thermally activated processes. The different values of A(T ) obtained from Figure 3
are plotted as a function of 1/T in Figure 4. From the slopeof the solid line, we have determined the activation energy
W defined byA(T ) = A0 exp(−W/kB T ),(1)

where kB is the Boltzmann constant. The obtained value isW ≈ 65kB TR , with TR = 339 K. This value of the activation energy is comparable to those reported in reference [9]both for thermotropic and lyotropic lamellar phases. (Itshould be noted that, in Ref. [9], a dilute lyotropic system
of CpCl/hexanol/brine exhibits a completely different exponent; m ≈ 5. In this case, the activation energy is muchsmaller.) The physical interpretation of the obtained activation energy will be discussed in Section 5.
For fixed T = 66 ◦ C, we varied the composition φ andrepeated the same measurements. The result is plotted in
Figure 5 for four different compositions. Although a powerlaw behavior is observed for each composition, the exponent m actually varies between different compositions. In
Figure 6, we plotted the obtained exponents as a functionof the composition, together with the theoretical predictions by Meyer et al. (m = 5/3) [9,10] and by us (m = 3/2,
as described in the next Section). In general, the observed

45 wt%
40
35

0.1

30
0.1

1
σ [Pa]

Fig. 5. (Color online) Log-log plot of the steady-state shear
rate γ̇ as a function of the shear stress σ at different compositions but with fixed temperature T = 66 ◦ C. The slope of thestraight lines differs slightly among different compositions.

exponent is close, but slightly smaller than our prediction m = 3/2. The deviation from m = 3/2 is largest
when φ = 45 wt% compared to the other three compositions. This result would be consistent with the observed
difference in the defective textures as we shall present in
Section 4. The difference may be due to a gradual structural transition in the sample, leading to a weaker shear
thinning behavior. Although it is known that a dynamical transition to the so-called onion structure occurs in
the presence of a relatively large shear flow [14,15], we
consider that it is not the case for φ = 45 wt%. This is because the exponent m would be much larger in the onion

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1.8

Table 2. The values of the stress scaling exponent m definedby γ̇ ∼ σ m for different systems.
SystemCTAB/hexanol/water
SDS/pentanol/dodecane
8CB (31 ◦ C)
CpCl/hexanol/brine
8CB (22–29 ◦ C)
P123/butanol/brine
C10 E3 /D2 O
C12 E5 /water

m

1.6

1.4

1.2

1.0

30

35

40

45

φ [wt%]
Fig. 6. The exponent m as a function of the composition.
The solid line is our theoretical prediction m = 3/2 while thedashed line corresponds to m = 5/3 which is predicted by
Meyer et al. in references [9, 10].

phase, and the texture of the φ = 45 wt% sample (see later
Fig. 12(d)) does not indicate any onion formation.
Table 2 summarizes the reported values of m in theliterature for various smectic or lamellar phases including
both thermotropic and lyotropic cases. The temperature
range is specified for a thermotropic liquid crystal 8CB reported in different papers. When the system undergoes astructural transition, we picked up the exponent that characterizes the weak shear regime. Here m typically ranges
between 1 and 2 but its precise value is not yet determined.

3 Theory
3.1 Dislocation loop under shear flow
In this section we present our scaling argument on defect dynamics, which predicts the exponent m = 3/2 as
mentioned above. The screw defects in lamellar or smectic phases cost very small free energy compared with theedge defects. Therefore the tangent direction of the screw
defects will remain perpendicular to the layers even in thepresence of a gentle shear. The two extremities of a screw
defect line will experience different ambient velocity under
the shear flow. If the two edges linked to the two ends ofthe screw have their tangents perpendicular to the flow
velocity (pointing along the vorticity direction), the flow
will have different relative velocity to the two edges. This
relative flow introduces the dissipation similar to the situation of edge climbing. Following Oswald and Allain [18],we regard this as the major dissipation. The viscous stresstherefore depends on the defect density.
To evaluate the defect density at a given shear rate,
we will estimate the defect production rate and the annihilation rate separately. The effects of deformation on
the defect production in crystal solids have been discussed

m
2
1
2
1.67
1.67
1.67
2
1.44

Reference
Bohlin et al. [13]Roux et al. [14]Panizza et al. [15]
Meyer et al. [8–10]
Meyer et al. [9, 10]
Meyer et al. [10]Oliviero et al. [16]present study

by Bruinsma et al. who predicted the shear-induced defect proliferation [19]. In that picture, the deformation enhances the dissociating rate of the bound pairs of defectlines, which then dissociate and move to reduce the applied strain.
In lamellar structures, many equilibrium defects arebound loops consisting of both screw and edge dislocations, so that the above mechanism should still operate.
However, lamellar phases have a well-known dilation (orundulation) instability. This instability can generate far
more defects than to dissociate the existing bound loops.
The question is, why does instability happen during theflow? If the initial layers are already tilted so that the layer
normal has a non-vanishing positive component along theflow direction, then the shear flow will tilt the layer normal
toward the flow direction. The shear will also dilate thelayer repeat distance. Therefore with the existing tilt, thelayers are dilated and the tilting angle is increased by theshear continuously, so that when the tilt reaches a certain
angle, the instability sets off and the layers are buckled.
This reasoning, however, fails when the sample starts
with the homeotropic alignment, where the shear cannot
tilt the layers as the layers are parallel to the local flow
velocity. In the next subsection, we suggest that, when theapplied shear stress drives the screw defects motion due
to the Peach-Koehler force, these defect motions will tilt
the layers, even when the layers are well aligned at the
beginning. Once the layers are not at the perfect alignment, the shear flow can tilt and dilate it as mentioned
above. The shear can now trigger the dilation instability to
produce new defect lines. Thus the dislocation productionin the lamellar liquid crystals is fundamentally different
from that of the crystals. In the last part of this section,
we estimate the (shear rate dependent) density and sizeof defect loops in the steady-shear condition. The defectline density is then related to the stress through the dissipation argument. The obtained shear thinning exponentm = 3/2 is found to be consistent with our experimental
result as described in the previous section.
Among the possible configurations of line defects andwalls, we will only consider the effect of the dislocation defect lines for simplicity. In fact, small-angle grainboundaries can also be viewed as an array of strongly
bound line defects.

C.-Y.D. Lu et al.: Non-linear rheology of lamellar liquid crystals

3.2 Defect motion and the layer tilting

Edges

Consider a lamellar sample aligned in the homeotropic orientation which is not tilted initially. The lamellar contains
some defect lines. A shear is applied such that the layer
normal unit vector n is in the gradient direction ẑ to begin with. The defect lines are assumed to form dislocation
loops, each consisting of two segments of screw dislocations (of length ls ) in the layer normal direction ẑ andtwo edge dislocations (of length le ) within the velocityvorticity plane.
Under the applied stress σ, the defect lines are subjected to the Peach-Koehler force per unit length [2,20,21]
FPK = τ × (σ · n b),(2)where n b is Burgers’ vector, and τ the unit tangent vector of the defect line. To define Burgers’ vector, one encloses the defect line by a closed loop where the net layer
distortion after completing the loop defines Burgers’ vector [2]. The equation transforms properly if one changes
the signs of n b and τ simultaneously. The stress σ does
not include the self-stress from the defect itself [2]. The
tension force ft from two ends of a defect line segment
gives a net force per unit length Ft = ∂ft /∂s, wheres is the arc-length of the defect line. The viscous force
is Fv = µ−1 (τ ) · (v(R(s, t)) − Ṙ), where µ(τ ) is theorientation-dependent mobility matrix, R(s, t) the position of the defect line, and v(R(s, t)) the convection velocity at R(s, t). The arc-length variable s increases along
the local tangent vector τ , so that τ = ∂R(s, t)/∂s. The
force balanced condition
FPK + Ft + Fv = 0(3)gives the defect line dynamic equation such thatṘ = v(R(s, t)) + µ(τ ) · (τ × (σ · n b) + ∂ft /∂s).

(4)

Note that for a given loop geometry R(s, t), there are two
possible defect orientations (depending on the sign of b),
which move differently under the shear.
The mobility matrix µ(τ ) depends strongly on the orientation and Burgers’ vector. For screws where τ = ±n,
the mobility is isotropic so thatµ(n) = µs (I − τ τ ),(5)

where I is the identity matrix, and µs the screw mobility.
The screws could move easily so that µs is large [21]. The
term −τ τ is to emphasize that the line will not movealong its tangent direction.
For the edge dislocation where τ ⊥ n, the mobility isµ(τ ) = µg nn + µe (I − nn − τ τ ),

95

(6)

where µg and µe are the edge glide and climb mobilities,
respectively. The latter can be measured, by using a wedge
to induce the layer dilation. The climb motion is slower,
being limited by the very dissipative permeation process.
The edge glide mobility µg is even smaller than µe because

Edges
ScrewsMotion under shear

Shear

Fig. 7. The screw defects move under the applied shear, toincrease or decrease the area of the defect loops.

gliding requires an activation process to initiate the layer
reconnection. Below we will neglect the edge gliding andset µg ≃ 0.
According to equation (4), the shear stress and its
associated Peach-Koehler force induce the motion of thescrews toward the vorticity direction. (The tilting effects
on screws due to the shear convecting flow are most likely
to be canceled by the screw line tension resisting tilting.)
The two screws of a given closed loop will have opposite
tangent τ , and hence the most important consequenceis that they move in the opposite directions under the
Peach-Koehler force. The area enclosed by the loop will
either decrease or increase depending on the loop orientation, as shown in Figure 7.
We define the vector area per unit volume p(r, t) as(similar to Eq. (9.50) of Ref. [2] but with the extra 1/2
factor)Z
1X
bi Ri (si , t) × dRi (si , t) δ 3 (r − Ri (si , t)),p(r, t) =2 i(7)where i denotes the label for the defect loops. For example, the average p(r, t) for N square loops, each with anormal pointing in the flow direction x̂ and a size of ls × leinside a volume V , is (ls le N/V )x̂. The loop area is also
related to other ideas which consider the defect lines asa generalized “currents” [2] or “line charges” [22]. If one
writes dRi (si , t) in the above integral as τi dsi , then p(r, t)
can also be interpreted as the moment of the line charge
density bi τi δ 3 (r − Ri (si , t)). If the defect loop is on thevelocity gradient-vorticity plane, p will point in the flow
direction. In a well-aligned equilibrium sample, the defectloops of the two opposite orientations could have the same
total area but opposite contributions to p, so that the averaged p vanishes in equilibrium.
Following from equation (7), the moment p(r) evolves
as
Z h
i
1X
b
Ṙ × dR + R × dṘ δ 3 (r − R)ṗ(r, t) =2 i
−R × dR(Ṙ · ∇) δ 3 (r − R),(8)

where the subscript i is suppressed for clarity. The lastterm ∇δ 3 (r − R) will be dropped as it is averaged to zero.

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The second term can be rewritten, after integration byparts, as
Z
Z
∂ Ṙ
3
× R δ 3 (r − R) ds
R × dṘ δ (r − R) = −∂sZ∂(R δ 3 (r − R)) ds= Ṙ ×∂s
Z= Ṙ × dR δ 3 (r − R)Z
− Ṙ × R τ · ∇δ 3 (r − R) ds, (9)where the first contribution is the same as the first termin equation (8). The second contribution is averaged tozero and will be neglected below. Together we get
X Zṗ(r, t) ≃(10)
b Ṙ × dR δ 3 (r − R).
i

Since the pure screw defect lines have much smaller
energy than the tilted screw, the tensions force will keep
the screws parallel to the layer normal n. In such case, thetwo edges of a given loop must have the same velocity Ṙ,but they have opposite dR. Therefore, for the edges, the
term
X Z
(11)
b Ṙ × dR δ 3 (r − R)edges

is averaged to zero.
For the screws, their dR is parallel to n. Since, for agiven loop, there are two screws with opposite dR, thecontribution v(R(s, t)) in equation (4) for Ṙ is the samefor the two segments with opposite directions, so that theircontributions average to zero. The tension term in equation (4) will shrink the loop areas for both types of loop
orientations. If the screw densities are the same for bothtypes of orientations, the net effect on ṗ also vanishes
on average. Therefore the remaining Peach-Koehler force
term will give the main contributionṗ(r, t) = K · (σ · n).

(12)

Here the tensor K is isotropic K ≡ (I − nn)K, withX Z
b2 µs δ 3 (r − R(s, t)) ds = hb2 µs iρs , (13)K=screws

where
ρs =

X Z

screwsδ 3 (r − R(s)) ds(14)

is the screw density, and the angular bracket denotes theaverage over the defect lines. Physically, ρs is the surface
number density of the crossings made by screw dislocations on a surface with its normal in the n direction. If
one regards the defect as a vector charge, K consists of themultiplication of the charge density, mobility, and chargesquared. The formula is similar to the one of the conductivity of the electrolyte solution so that K can be viewed
as a generalized conductance tensor.

To describe the layer orientation and the layer compression, one definesm = (d0 /d)n(15)

as the scaled wave vector variable, where d is the local
distance between the two neighboring layers, and d0 theequilibrium layer separation. Recall that m represents the
local layer compression, so that the integration along an
arbitrary loop defines Burgers’ vectors of the defect lines
enclosed by the loop. In a differential form, it reads
XZbi τi δ 3 (r − Ri (si , t)) dsi ,(16)∇ × m(r, t) =i

as in equation (9.18) of reference [2].
Now it can be demonstrated how the change of defectloops area would cause the change of the average layer
orientation by establishing the relationship between p andm. Using the identity ǫabc rb ǫcde ∇d me = 2ma +∇a (rb mb )−∇b (rb ma ), we writeZZ
1ǫabc rb ǫcde ∇d me drma (r, t) dr =
2 V
V
Z
1
+
(ma rb − mc rc δab ) dAb
2 ∂VZ=pa (r, t) drVZ1(ma rb − mc rc δab ) dAb , (17)+2 ∂Vwhere the term ǫabc rb ǫcde ∇d me (or r × ∇ × m) is transformed into p(r, t) as
XZ
1
1
r×∇×m = r×bτ δ 3 (r − R(s, t)) ds
2
2
Zi
1X
R(s, t) × bτ δ 3 (r − R(s, t)) ds=
2 i= p(r, t).

(18)

Equation (17) shows that the perturbation of the vector
field δp will induce δm. The Peach-Koehler force under
the shear will induce δp along the velocity direction, whichis perpendicular to the initial m. Therefore the perturbation of the defect loop area (due to the screw motion under
the shear), can cause the layer orientation (the m direction) to tilt toward the flow direction. We finally obtain
the evolution of m asṁPK = K · (σ · n).

(19)

The important consequence is that, even starting froma strictly homeotropic aligned sample, equation (19) gives
a ṁPK in the flow direction, leading to the layer tilting
driven by the defect motions. A graphic illustration isshown in Figure 8. Furthermore the direction of tilting is
the unstable one with which the shear flow will further increase the tilting toward the undulation instability. This

C.-Y.D. Lu et al.: Non-linear rheology of lamellar liquid crystalsab

tilting bydefectmotionsdFig. 8. Time sequence drawings to illustrate the layer tilting
caused by the defect loop motion. The large arrows indicate theshear flow. The small arrows indicate the screw defect velocity.
(a) The defects move under the applied shear. (b) As the screws
move, the loop areas increase or decrease according to the loop
orientations. (c) As a consequence of the screw defect motion,
the layers are tilted.

sequence is illustrated in Figure 9a–c. Researchers haveobserved layer dilation under flow constantly. However, it
is not obvious that a well-aligned layered structure withhomeotropic orientation should encounter layer dilation.
(Of course the slopped boundary plates may also cause
dilation as the gap changes under shear.) Here we show
that, as long as the layered structure has defect loops, itbounds to be tilted and dilated under the applied stress.
The dilation eventually leads to undulation instability
and defect formations, with which the homeotropic andpreferred layer spacing could be maintained. This undulation instability at the non-linear regime was indeed observed to produce edge dislocations parallel to the undulations [23,24]. Usually the defect formation involves thebreaking of the layers, hence the outcomes are relatively
noisy. The lamellar or smectic phases, however, show surprisingly regular pattern upon dilation instability. Thissuggests that the defect formation scheme might not have
many variations. The mechanism could be close to the picture illustrated in Figure 9c–f. As shown in the sketch, thedilation caused by the shear flow eventually leads to instabilities and a pair of edge dislocations are produced suchthat the layers could align horizontal and restore theirpreferred layer spacings. In a real three-dimensional sample, the two edges are connected by two screws to forma dislocation loop. The orientation of these newly formed
defect loops must be predominantly the one disfavored by
the shear, so that m, and hence the layer normal n, can

e

97

c

tilting &dilation by
convectionfFig. 9. Sequences of layer titling by defect motions,convection-induced dilations, and defect formations. a→b: layers tilted by the changes of defect loop area due to screw motion(defect lines not shown for clarity). b→c: layers further tilted
and dilated by convection. c→e: onset and development of theundulation instability. e→f: creation of new defect loop, withthe edge segments running perpendicular to the paper and thescrew segments (not shown) running vertically.

be restored to the homeotropic alignment, as illustrated
in Figure 9f. This realignment to homeotropic thus gives
an estimate of the minimal production rate of the screw
defects by the instability mechanism.
Since the layers are first tilted by the defect motions
and subsequently dilated by the shear flow, we thus postulate that, as |m| is decreasing with layer tilting, undulation instability happens at a particular tilting angle
θc [25]. The production of the defect loops takes place at afrequency proportional to γ̇. Experimentally, the fluctuating characteristics of such a sample might be similar to theoscillation in jamming of a colloid [26]. In the colloid system, it is indeed observed that the stress fluctuates with acharacteristic time proportional to the inverse shear rate.
3.3 Screw density and the shear stress
Next we calculate the viscosity contributed by the defectloops. When the layers are sheared with a rate γ̇, the defect loops could cause energy dissipation, mainly from theviscous force acting on edges by the shear flow. Let us
assume a typical dislocation loop with screw lengths ls
and edge lengths le as defined before. We follow the argument by Oswald and Allain to estimate the dissipationfrom the viscous force [18]. The argument is also consistent
with the equation (4) above. Each defect loop floats in theshear flow as a whole, so that Ṙ becomes the same everywhere on the loop. However, the two edges are advected by
the flow with the different velocities v(R(s, t)). Therefore
the edges have to climb (driven by µ · ∂ft /∂s). The climb

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Combining equations (20) and (22), the stress σ = ηeff γ̇
becomes
ls le γ̇σ ≃ ρs≃ ρs τe ,(23)µewhere we have used the identity ρs = ls n0 .
3.4 Shear thinning behavior
Fig. 10. Elongation of defect loops when the viscous force
exceeds the edge line energy.

velocity has magnitude |Ṙ − v(R(s, t))| = ∆v = ls γ̇/2
for both edges. The viscous force on the edge is therefore
fv = le |Fv | = le ∆v/µe . The dissipation per unit volume
T ṡ gives the scaling of the effective viscosity ηeff as
 2 
n0 ls le2
T ṡ ≃ ηeff γ̇ ≃ n0 ∆vfv ≃γ̇ 2 ,(20)µe
where n0 is the number density of the loops.
To balance the desired viscous force, where Ṙ is thesame for the loop, the required tension fs is
fs ≃ls le γ̇
.
µe

(21)

The actual tension has magnitude le |Ft | ≃ τe which is theenergy per unit length of the edge dislocation. If fs > τe ,the actual tension is not strong enough to maintain theconstant Ṙ for both edges, and two new edge segments
along the flow direction will be produced, as illustrated
in Figure 10. This leads to a continuous elongation in
the flow direction. Such deformed loops may collide withother defect lines, causing defect annihilation and reducing loop sizes.
Right after the undulation instability, we thus expect
the formation of defect loops with long edges. However
they will encounter huge viscous forces on the screws ifthey were able to maintain the stationary shape. The actual tension is unable to match the magnitude. Therefore the large loops only appear in transient, i.e., theymust be reconnected or pair-annihilated. The pair annihilation requires matched Burgers’ vectors and hence happens slowly. We assume that during the short transientperiod, the large loops collide and reconnect to form manysmall loops, so that after the reconnection, the defect loops
are small enough to reach the balance fs ≃ τe . We assume
that the cascade of defect reactions eventually constraints
the defect loop areas so thatls le γ̇≃ τe ,µe(22)i.e., the line energy is able to maintain the loop geometry
in the shear flow. A similar steady-state consideration ofwhether the loop will be tilted by the flow leads to a similar scaling, with τe replaced with the screw line tension.

We then consider how the shear rate affects the defectdensity. Suppose that the instability happens at a particular tilting angle θc and that the instability creates certain
new defect loops, of density ∆n0 . The loop area produced∆n0 ls le must be at least θc /b, so that the layers are tilted
back (to the homeotropic alignment). In other words, thisis a lower bound estimation and any lower defect production rate will not be able to maintain the area balance
between the two types of loop orientations. The characteristic time between two instabilities is ∆t = θc /γ̇. Recall
that ∆ρs = ls ∆n0 , which leads to the production rate of
ρs as
∆ρs
γ̇
=
.
(24)
∆t
ble
These rectangular loops are convected by the shear flow.
For a particular loop, there is a distance ξ in the flow
direction to the next loop whose central position is shiftedvertically not larger than ls (so that they have a chanceof colliding). Since two colliding loops on average havea vertical position separation of ls , their relative velocity
under shear flow is γ̇ls . The colliding rate is thus γ̇ls /ξ
and the complete rate equation for ρs is
γ̇
γ̇ls
∂ρs
=
−
ρs .
∂t
ble
ξ

(25)

With ρs ≃ ξ −2 , we obtain in the steady state as
ρs ≃



1
bls le

2/3

≃



γ̇
bµe τe

2/3

,

(26)

where we have used equation (22). The scaling exponentof 2/3 on the shear rate γ̇ originates from the fact thatthe loop number density increases with γ̇ but, on the other
hand, the loop size decreases with γ̇ because of the limited
line tension support. From equation (23), we obtain thescaling relationbµe(27)γ̇ ≃ 1/2 σ 3/2 .
τeHence we get the exponent m = 3/2 as mentioned in Section 2 and the rheology response is a shear thinning behavior.
The line energy of the edge defect contains the elastic
energy contribution, which is sensitive to the defect core
structure. The integrated energy depends on the shortdistance cut-off as detailed in reference [2]. Whenp Burgers’ vector is smaller than the penetration length Ks /B,
where Ks is the saddle-splay elastic constant and B thelayer compression modulus, we use the penetration length

C.-Y.D. Lu et al.: Non-linear rheology of lamellar liquid crystals

as the cut-off to estimate τe ∼ Bb2 . Then the above relation is further simplified asγ̇ ≃µe 3/2σ ,B 1/2µe 3/2b1/2
σ .
(Ks /B)1/4 B 1/2

(a)

(b)

(c)

(d)

(28)

which depends only on µe and B, and is independent of b.
When the averaged
p Burgers’ vector is larger than the penetration length Ks /B, the cut-off is set√
to Burgers’ vector, and the edge energy becomes τe ∼ b Ks B. We thenobtain instead
γ̇ ≃

99

(29)

The obtained shear thinning exponent m = 3/2 iscompared with our experimental result in Figure 6 together with the prediction by Meyer et al. m = 5/3 [9,10].
Other experimental values of m are listed in Table 2. They
scatter between 1 and 2. Interestingly, the material 8CB
measured at a slightly different temperature shows clearly
different exponents [10,15]. The exponent m = 2 is reported for the higher temperature where the system isjust a few degree lower than the smectic-nematic transition point [15]. The smaller exponent m = 1.67 is reported
for a lower temperature sample [10]. It will be certainly
helpful to have more systematic study on the lamellar andsmectic rheology behavior.
If we put some typical numbers into equation (28) suchas µe ≃ 102 m s kg−1 , B ≃ 106 Pa [27], σ ≃ 0.5 Pa, we
obtain γ̇ ≃ 0.035 s−1 . This value is slightly smaller thanthe measured shear rate γ̇ ≈ 0.5 s−1 but gives roughlya reasonable order. Another estimate of B ≃ kB T /d30 ≃104 Pa would yield γ̇ ≃ 0.35 s−1 . One possibility for the
low value is that the sample has larger Burgers’ vectors,
so that one should use equation (29) instead. Using thevalue b ≃ 2 nm [18], and the above shear rate, we get the
average distance between screw dislocation ξ to be roughly
2–3 µm. The order of magnitude for ξ matches the distanceseen in the previous experiment [10].

4 Texture visualization
We also performed some texture visualization of our samples. For the optical crossed-polars microscopy observation
under shear flow, we used CSS-450 from Linkam Scientific
Instruments, Ltd. The gap between the two plates was
200 µm. The pictures were taken at constant shear rates.
In Figures 11 and 12, we show the observed micrographs under shear flow. The shear rate is γ̇ = 0.5 s−1
for φ = 35 wt%, and γ̇ = 1 s−1 for φ = 45 wt%. For
φ = 45 wt% sample, the elongated objects oriented along
the flow in the steady state are typical oily streaks. The
structure observed here is similar to that of Meyer et al.
for 8CB [10].
For φ = 35 wt% sample, the elongated objects oriented
with a small angle with respect to the flow in the steadystate. The structure is somewhat different from the typical oily streaks. The elongated objects seem to be densely

Fig. 11. Optical crossed polars micrographs under shear flow.
The concentration of C12 E5 is φ = 35 wt% and the shear rateis γ̇ = 0.5 s−1 . (a) 0 min, (b) 3 min, (c) 5 min, (d) 10 min. The
scale bar in (a) is 100 µm and the flow direction is indicated byan arrow in (d). The flow gradient direction is perpendicular
to the paper.

(a)

(b)

(c)

(d)

Fig. 12. Optical crossed polars micrographs under shear flow.
The concentration of C12 E5 is φ = 45 wt% and the shear rateis γ̇ = 1 s−1 . (a) 0 min, (b) 5 min, (c) 10 min, (d) 15 min. The
scale bar in (a) is 100 µm and the flow direction is indicatedby an arrow in (d).

packed thin bright lines of few hundred microns in length.
There are dark regions at which the bright lines terminate.
The dark regions tend to align along the direction perpendicular to the thin lines. We are unable to fully explain
the observed texture. More study in the future will help
the detailed modeling. It is possible that these bright lines
are densely packed, weakly contrasted oily streaks, and thedark regions contain screw defects. This is because the oily
streaks come from the partial merging between two edge
defects with Burgers’ vectors of opposite signs. (The fully
merged two edges will become a simple edge.) The φ = 35
and 45 wt% might have oily streaks with different extent
of merging, which affects optical appearance.

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For φ = 30 and 40 wt% samples, the structures are
similar to that of φ = 35 wt% sample. In the φ = 40 wt%
sample, we also see that some typical oily streaks appear
together with the thin lines (presumably edge defects) ina parallel way. As the system reaches steady state, thenumber of typical oily streaks decreases.
The defect textures correlate strongly with the rheological shear thinning exponent. At smaller φ the observed
exponents are close to the theoretical prediction, albeit theobserved textures are more complex than the simple edge
defects modeled in the theory. The theory proposed above
does not work well for φ = 45 wt%, and there the textures
are different from the ones of other concentrations.

5 Discussion
Several points merit further discussion. The theory proposed assumes that the interaction between the screw defects is much smaller than the Peach-Koehler force. This
includes the line-tension–induced attraction between thetwo screws connected by the edges, and the direct interaction between the defect lines. In the optical microscopevisualization, we observe the fine texture with dark lines,
probably indicating the cluster of the defect lines. It is notclear how the defect interaction affects the rheology behavior. Nonetheless, the measured shear exponent is not too
far from the proposed simple picture which does not take
into account defect interaction. This may indicate that thedefect density scaling is close to the one estimated here.
However, our estimations for the defect production andthe annihilation rates are both too high. Within the allowed time window, no defect texture coarsening happens
once they reach the steady-state density. We are unable toidentify any sign of a dynamic balance between the defectproduction and the defect annihilation.
Next we discuss the similarities and the differences
between the present theory and the one by Meyer et
al. [9,10]. Similar assumptions for both theories are: i) The
two theories both assume that the screw defects move according to the Peach-Koehler force. ii) Both theories consider that the plastic deformation is due to defect motion.
The plastic layer tilting rate here (see Eq. (19)) is similar to the combination of the Peach-Koehler force and the
Orowan relation there.
However there are important differences which leadto different mechanisms for rheology responses: i) In thepresent theory we predict that the plastic deformation
will lead to instability, as the layer tilting driven by the
Peach-Koehler force is toward the unstable direction. In
the theory of references [9,10], they concluded that thesign of the plastic deformation helps to stabilize the layer.
Therefore even under the shear flow deformation whichleads to instability for a tilted layer, their theory assumes
that the instability does not happen. ii) In our theory,the plastic deformation is only important initially, whichsets up the initial layer tilting. After the initial tilting, we
assume that the shear deformation dominates. In the theory of references [9,10], the plastic deformation rate is thesame as the flow deformation rate all the time. iii) The

defect line density is determined differently. In their theory the applied stress is mainly balanced by the internal stress, which comes from the interaction between thescrew defect lines. This condition fixes the screw density
in references [9,10]. In the present theory we ignore theinteraction between the screws. The defect density here
is determined by the balance between the production andannihilation processes as in equation (25). iv) The defectorganizations are assumed differently. We assume that thedefect interaction is screened so that the defect is free tomove under the applied force; whereas in their theory thedefect lines interaction is important, which provides theinternal stress under applied stress. Nevertheless, the defect line can still move. In other words, we consider thedefects as in the “liquid state”, whereas they seem to view
defects as in the “glass state” which has the shear modulus due to defect interaction. v) Two theories have considered different dissipation mechanisms. The dissipation
here is due to the permeation which happens when themass flow passes through the edge defects. In their theory the external force has to do work against the internal
stress to move the defects. If these two mechanisms areboth important, the sum of their dissipation should beconsidered. vi) As pointed out in iii), the stress in references [9,10] has an elastic nature, whereas in our theory
the stress is assumed to have viscous origin. Further stressrelaxation experiments after the shear rate jump may give
useful information to identify the nature of the stress.
In Section 2, we have measured a relatively large activation energy, 65kB TR . The large activation energy inthis system indicates a collective process involved in theflowing condition. The shear flow imposed on a standing
defect loop causes an edge climb motion relative to its
local environment. The climb of an edge requires the surfactant to diffuse to or away from the layers containing
the defect edges. This diffusion, or permeation, has a flux
normal to the layer. When the permeation flux involved an
activation process, the permeation coefficient contains theactivation energy. Three possibilities come to mind. i) The
first possible path requires the surfactant to dissolve into
the water, diffuse to the other layer, and readsorb into thebilayer. If that is the dominant path, then the activationenergy corresponds to the energy barriers for a surfactant
molecule to dissolve or readsorb into the bilayers. One expects that the activation energy comes from the van der
Waals attraction or the hydrophobic/hydrophilic energy.
The experimental value, however, seems much too large
for this single molecule interpretation. ii) The second possible pathway is that the dissolving process involves morethan one surfactant molecule, to act collectively. The dissolved molecules may form a small micelle directly. This
collective process may be similar to the vesicle budding
process in lipid bilayers. iii) The third possible pathway
for the permeation flux is that the adjacent bilayer canform a neck to fuse and then break up, which facilitates
the exchange of molecules between the layers. If the secondor the third collective processes have a higher rate than
the single molecule process, they may dominate the viscosity and become very sensitive to the temperature. The
third process is similar to the relaxation of the surfactant

C.-Y.D. Lu et al.: Non-linear rheology of lamellar liquid crystals

sponge phase, where the neck forming and breaking relaxation can be monitored by neutron scattering [28], andthe activation energy can also be measured.

6 Conclusion
The shear thinning behavior is measured for the lamellar phase of the C12 E5 /water system. The shear thinning
exponent depends on the concentration. A simple rheology theory of lamellar phase is proposed with the relationγ̇ ∼ σ m , with m = 3/2. In the theory, we have shownthat the defect-mediated layer tilting and dilation undershear flow could cause continuous production of dislocations. This picture explains the source of the defects. For
concentration around 30 wt%, the shear thinning exponent is close to our theoretical estimation. Whereas at ahigher concentration, close to 45 wt%, the exponent becomes smaller. The optical texture observation indicatesthat the texture changes very much with the concentration. This confirms that the defect texture could be responsible for the rheological behavior.
This work started when CYDL was visiting KITP at UCSB.
He also acknowledges support by the National Science Councilof Taiwan. SK and TK acknowledge support by KAKENHI(Grant-in-Aid for Scientific Research) on Priority Areas “SoftMatter Physics” and Grant No. 18540410 from the Ministry of
Education, Culture, Sports, Science and Technology of Japan.
We thank Prof. M. Imai for his help on the experimental partof the present work.

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