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Zigzag disclination in uniaxial nematic phases : study in capillary tubes
Martine Mihailovic, Patrick Oswald
To cite this version:
Martine Mihailovic, Patrick Oswald. Zigzag disclination in uniaxial nematic phases : study in capillary tubes. Journal de Physique, 1988, 49 (8), pp.1467-1475. �10.1051/jphys:019880049080146700�. �jpa- 00210827�
Zigzag disclination in uniaxial nematic phases : study in capillary tubes
Martine Mihailovic (1,2) and Patrick Oswald (1)
(1) Laboratoire de Physique des Solides, Université de Paris-Sud, Bât. 510, 91405 Orsay Cedex, France (2) Laboratoire des Solides Irradiés, Ecole Polytechnique, 91128 Palaiseau Cedex, France
(Reçu le 9 f6vrier 1988, accepté le 25 avril 1988)
Résumé. 2014 Nous avons observé des lignes de disinclinaison en zigzag dans des nématiques uniaxes thermotropes et lyotropes. Nos échantillons sont préparés dans des tubes capillaires ronds et rectangulaires.
Cette géométrie confinée et l’instabilité des lignes vis-à-vis de perturbations tridimensionnelles sont
responsables de la forme observée.
Abstract. 2014 We have observed zigzag disclination lines in uniaxial thermotropic and lyotropic nematic liquid crystals. Our samples are placed in round and flat capillary tubes. This confined geometry and the instability of
the lines under three-dimensional perturbations are responsible for the observed shape.
Classification
Physics Abstracts
61.30 - 61.70G - 61.70J
1. Introduction.
Disclination lines are characteristic defects of the nematic phases. On these lines, the orientational order of the molecules is broken. These phases are
fluid so the lines often have a rounded shape. Seen through the polarizing microscope they look like
flexible threads : leading thus to the name nematic,
nema meaning hair in greek.
Recently, Galeme et al. [1] have seen in a biaxial lyotropic nematic a disclination making spon-
taneously a zigzag which disappears at the biaxial
nematic --> uniaxial nematic transition. It could be
suggested that the biaxiality alone is responsible for
this shape. Actually, zigzag lines exist in ordinary
uniaxial nematic: biaxiality cannot be proved by
such an observation.
To see a zigzag line, one can look at the boundary
between two nematic domains of different orien- tations. This method has been described recently by
Galeme et al. [2]. Besides, under peculiar conditions,
round capillary tubes filled with a nematic phase can
exhibit a pair of zigzag lines along their axis. This has been already noticed but not explained by Cladis [3].
We repeat this experiment and do a new one in a
semi-circular geometry and then we analyse the zigzag instability both experimentally and theoreti- cally.
2. Observations in capillary tubes.
2.1 SAMPLES. - A lyotropic liquid crystal composed
of 43.3 wt % DACI (decyl ammonium chloride)
52.35 wt % H20 and 4.33 wt % NH4C’ (mixture
called D3) has been studied. The phase transitions
are :
The samples are put into flat capillary tubes (thickness : 20, 50, 100, 200, 300 )JLm) (Fig. 1) and in
round capillary tubes (diameter : 50, 100, 150 um)
from Vitrodynamics INC. The tubes are immersed in optical oil of same index as the glass, set into a
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019880049080146700
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Fig. 1. - Schematic representation of a flat capillary
tube.
Mettler hot stage and observed through a polarizing microscope.
2.2 OBSERVATIONS IN FLAT CAPILLARY TUBES.
2.2.1 Lyotropic system D3. - Extinction of the light
between crossed polarisers shows that the layers in
the lamellar phase naturally lie parallel to the flat
surfaces of the tube (Pl on Fig. 1). Such an orien-
tation is called homeotropic (optical axis normal to
Pl). Focal conics are only visible near both lateral
sides of the tube (Fig. 2a). On warming into the
nematic phase (above 37.5 °C) the focal conics
disappear and are replaced by a zigzag line (Fig. 2b,
c, d) located on both sides of the line L shown in
figure 1. This line separates the homeotropic region
from a birefringent one. This zigzag line persists up to the nematic-isotropic transition and can also be obtained by cooling the sample from the isotropic liquid. On warming, we observe qualitatively that
the wavelength A increases with temperature and is
proportional to the thickness 2 R (Fig. 3) of the sample while Q (angle between the zigzag and the
direction L, Fig. 1) is approximately independent of
these two quantities.
Fig. 2. - Observation between crossed polarizer and analyser of the D3 mixture in flat capillary tubes. (a) Lamellar phase. Focal conics persist near the semi-circular edge of the tube. The sample is homeotropic elsewhere. Thickness : 2 R = 50 um. (b), (c) Nematic phase : two pictures of the same region. A zigzag line develops near the edge of the tube.
2 R = 100 lim. (d) Nematic phase. 2 R = 200 um. Note that the wavelength scales like the thickness.
Fig. 3. - Angle cp of the zigzag line versus the thickness of the sample (mixture D3).
Because of the weak birefringence of the mixture D3 and typical problems of lyotropic systems such as drying and inhomogeneity of concentrations, it is difficult to make any accurate measurement of A and
(p versus temperature.
2.2.2 Thermotropic nematics (5CB, 6CB, 8CB). -
In order to get homeotropic anchoring, the glass is
coated with a silane (ZL1 3124 Merck). In each case
a zigzag line is observed near both sides of the tube in the nematic phase (Fig. 4). Measurements of A and cp are reported in figure 5 for 8CB. Their
evolution versus temperature and thickness of the sample is qualitatively the same as in D3.
Fig. 5. - Wavelength A and angle cp of the zigzag in 8CB
versus AT = T - Tc (Tc = Tsm - N ).
Fig. 4. - Observation between crossed polarizer and analyser of 8CB in rectangle tubes. (a), (b), (c) 2 R =
20 um, AT = 0.05, 0.15, 0.30 °C respectively.
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We tried to look at the behaviour of À and Q close 2.3 OBSERVATIONS IN ROUND CAPILLARY TUBES.
to the smectic-nematic transition. Because of the We have to distinguish between compounds
poor resolution of our oven, it was impossible to which undergo a smectic-nematic transition (8CB, approach the transition better than 0.05 °C. Never- D3) and those which have only a nematic phase
theless we were able to see the effect of a very small (5CB, 6CB). Homeotropic anchoring is again
thermal gradient along the capillary tube : when the favoured by using silane.
wavelength saturates at about 1.2 R, the angle Q
vanishes at the smectic -->. nematic transition. On the 5CB and 6CB behaviour is rather well known : a
contrary, A and Q remained both finite at the line without singular core is along the axis of the nematic - isotropic transition. capillary tube (Fig. 6a). Point defects can exist on
Fig. 6. - Observations in round capillary tubes between crossed nicols. (a) 8CB : disclination line without singular
.core. 2 R = 150 um, AT = 0.5 °C. (b) 8CB : sample in a temperature gradient. From the left hand side (smectic phase)
to the right hand side (nematic phase) the temperature increases of about 0.1 °C. The central line (which is singular in
the smectic phase) splits into two singular lines in the nematic phase which repel each other. Both lines go progressively
over to an in-phase zigzag as the temperature increases. 2 R = 50 um. (c), (d) Same sample of 8CB at the same temperature. 2 R = 50 um, AT = 0.1 °C. Rotating the sample of 90° shows that the two lines are in the same diametral
plane. (e) Two singular lines can recombine to give way to a non singular one. 2 R = 150 um, AT = 0.5 °C.
this line, they have been described in detail pre-
viously by Cladis et al. [4].
The 8CB and the D3 behaviour depends on the
way we approach the nematic phase. By cooling
from the isotropic liquid we still observe a single line
with a diffuse core as in 5CB and 6CB. On the contrary, warming the sample from the smectic
phase leads to a splitting of the central line (which is
a singular one in the smectic phase) into two singular
lines repelling each other (Fig. 6b). As in rectangle tubes, both the li are unstable and change spontaneously into zigzags. Rotating the sample
around its axis (Fig. 6c, d) shows that the two lines
are in the same diametral plane. This is true only
near the smectic phase (AT = T - Ttransition 0.3 °C)
otherwise, both lines go out of this plane resulting in
a three-dimensional structure.
We measured A and cp dependence on the tempera-
ture and size of the tube (Fig. 7). It is almost the
same behaviour as in flat capillary tubes (see Fig. 5).
Although the two zigzag lines can coexist in the whole temperature range of the nematic phase, they
can also combine to give way to a non singular line (Fig. 6e). It means that the zigzag configuration is
metastable except near the smectic -> nematic tran- sition.
Fig. 7. - Wavelength A and angle Q versus temperature for 8CB in round capillary tubes.
3. Interpretation.
First we shall analyse the case of flat capillary tubes
and later the case of round capillary tubes.
For semi-circular geometry, topological consider-
ations will lead us to analyse a new molecular configuration of S
=1
disclination line where the molecules tend to align partly along the axis. Because of elastic anisotropy, this configuration is expectedto be less energetic than the one where molecules
remain in a plane perpendicular to the axis (planar line). Then we try to build a phenomenological
model of the zigzag instability.
For circular geometry we will have to explain the splitting of the S = 1 line at the smectic-nematic transition. The anisotropy of the elastic constants will be again very important.
3.1 S = 1/2 DISCLINATION LINE; GEOMETRICAL CONSIDERATIONS. - We look for neutral lines in
our sample by rotating together crossed polarizer
and analyser (Fig. 2b, c). On these lines, the director
(optical axis) or its projection onto the horizontal
plane PI is parallel or perpendicular to the analyser.
The director being normal everywhere to the glass boundary, it is possible to build a model of
S = 1 disclination line which agrees with these 2
optical observations.
Its main features are summarized below :
- in the horizontal plane PI which contains the disclination line, the field lines of the director are
normal to the glass and near the core, they become quite parallel to the line (Fig. 8a) ;
- when going from a zig to a zag the sign of the
curvature of the line changes. Then, there are singular points on the line itself matching the region
between a zig and a zag (Fig. 8a).
If 0 is the azimuthal angle, the curvature of the
field line (Fig. 8b) will be maximum for 0 = 0 (plane P1) and will vanish for 0 = 7r (plane P3) because
2
of the matching conditions with the homeotropic region.
Fig. 8. - (a) Field lines of the director in the plane Pi which contains the zigzag line. (b) Section by a vertical plane P2. A nail represents the director lying at an angle £
to the plane of the diagram. Its length is proportional to cos £ ; the head of the nail is below the plane of the figure
and its point is directed toward the observer.
In conclusion, the main feature of this S =
1 line is that the molecules tend to go out of the 2
plane normal to the axis of the line. A calculation of energy, taking into account elastic anisotropy, is
necessary to explain these observations.
1472
3.2 ENERGY OF A S = 1/2 LINE ; ROLE OF THE ELAS- TIC ENERGY. - In this section, we show that the disclination line where the director remains in a
plane perpendicular to the axis is not the most stable
one. To simplify our calculation, we make the following assumptions :
- the line is straight. The zigzag instability will
be analysed later on ;
- director field lines are arcs of circle (radius 3t)
centred on the glass boundary (homeotropic anchor- ing is then automatically satisfied) and lie in planes containing the axis of the capillary tube (plane 0 =
constant, Fig. 9) ;
-R is a function of 0. It is minimum for 0 = 0 (:R (o = 0) = 3to, 3to :::. R) and infinite for (J = :t 7T in order to achieve continuity with the ,
2
homeotropic region. One can notice that /3 = cos-1
( :0)
is the matching angle of the directorR0
field lines with the core of the disclination in the horizontal plane (Fig. 9).
Fig. 9. - Definition of :Ro and f3 in the horizontal plane P1.
The total energy per unit length is given by
where Rc is a core radius, n the director and
Ki, K2, K3 respectively the splay, twist and bend
elastic constants. To calculate explicitly this energy
R0
we choose 3t = ’Ro . . It yields in K2 unit :
Cos 0
where a is the ratio R ’Ro = cos 13 and K12 = K 1 K2 and
K3K32 = K2 .
In figure 10a, the splay, twist and bend contri- butions are plotted separately. Obviously, the bend
energy is small compared to the other two. Sub- sequently, we assume K32 =1. In figure 10b, we plot (AE)1 =E(a)-E(a = 0 ) for several values of the
anisotropy K12.
Note that E ( a = 0 ) is the energy of a line with radial field lines (13 == 2013 2 classical model of a planar line). These curves show that if K12 1.15 the
energy will be minimum for a = 0. This is the classical singular configuration. If on the contrary K12 > 1.15 the curve AE (a) will have a negative
minimum for a non zero value of a ( a = a * ) which
means that the previous configuration will not be
anymore the best one. In this case the molecules tend to align parallel at the core axis. Figure 11
shows the angle /3 * = cos- 1 a * which minimizes the energy as a function of the anisotropy K12. Its
variation is rapid when the anisotropy increases.
Fig. 10. - Energy of a S = 1 2 disclination line. (a) Splay (K1), twist (K2 ) and bend (K3 ) energy contributions as a function of a = cos 13 (R/Rc = 104) . E ( a ) = K1 E1 + K2 E2 + K3 E3. (b) Energy difference (AE)i=E(a)- E(a = 0 ) (in K2 unit) versus a for K32 = 1 and various values of K12-
Fig. 11. - Angle {3 * which minimizes the energy against
the anisotropy K12 for K32 =1.
K12 has been measured previously on 5CB, 6CB
and 8CB [5]. For 5CB and 6CB, K12 is always greater than the critical value 1.15. 8CB behaviour differs
only very close to the smectic-nematic transition where K2 diverges and K12 goes to zero. In con-
clusion, when the classical model is only valid for
8CB close to the smectic phase (AT 0.1 °C), for
AT:::. 0.1 °C, our model fits better because it con-
firms that the molecules go out of the radial plane.
3.3 ZIGZAG INSTABILITY. - Experimentally, it is
clear that the zigzag shape derives from the three- dimensional structure of the S = 1/2 line. The direc- tor escapes from the vertical plane P2 thus the only plane of symmetry of the configuration is P1. Then a driving torque on the line will necessarily be vertical and perpendicular to PI and the line will rotate in the
plane Pl. To understand qualitatively the origin of
this configurational torque one only needs to notice
that a small rotation of the line decreases the curvature radius of the field lines and thus the bend energy of the system (Fig. 12). Of course the defect
Fig. 12. - Rotating the disclination line decreases the radius of curvature of the director field lines in the horizontal plane P1.
line cannot go far from the axis of the tube because of the homeotropic anchoring on the glass. In other words, a return torque due to rigid boundary con-
ditions prevents the line from rotating. Besides,
inevitable singular points along the line will cost some energy too. In brief, zigzag instability is a
balance between a driving torque due to bend
curvature and a return torque due either to homeo-
tropic anchoring on the glass and to point defects
nucleation.
To go further, let us now evaluate the energy difference (per unit length along z) between the
zigzag configuration and a straight disclination line :
There are three contributions as previously discus-
sed :
- the first one is the bend energy which is saved
by tilting the line of an angle Q (driving torque). It
can be written in the form :
A (Kl, K2, K3 ) has the dimension of an elastic con-
stant ;
- the second one is due to singular points. It is
the energy of the matching region (of size R, see Fig. 8a) between a zig and a zag.
B(Kl, K2, K3, B ) has the dimension of an elastic
constant. Note that
B ({3 = ; ) = 0
2 (no singularpoints) ;
- the last contribution is due to the anchoring on
the glass (return torque). Assuming that each ele-
ment of the line experiences a force d f proportional
to its displacement z tg cp (d f = kz tg cp dz) we
calculate :
The spring constant is a function of R and Kl, K2, K3. The only possible combination is : k =
C where C(Kl, K2, K3) has the dimension of an
R2
elastic constant.
Finally, we obtain :
Minimization with respect to A leads to A =
B 1/3
R(
B l
1/3 and (åE)2 can be rewritten in theC tg2 cp
form (in A unit)
with
In this equation {3 and F are only functions of the elastic constants and thus of the temperature. Two
1474
cases at least must be considered, depending on the
value of the angle {3.
For f3 =; (planar line), (AE)2 (Q ) is always 2
positive so there is no instability as expected. This
situation is met experimentally near the smectic- nematic transition.
For {3 2" 2 and F small enough, (AE)2 ( cp )
vanishes and becomes negative so the zigzag insta- bility may occur. The slope being always infinite for
cp = 0, the curve (AE)2 ( cp ) has a positive maximum
as shown in figure 13. This maximum is due to the topological necessity of introducing singular points.
Then, the straight line configuration (Q = 0 ) is
metastable whereas the zigzag configuration (cp = cp * ) is stable. The corresponding wavelength
of the zigzag is given by :
Fig. 13. - Energy difference (AE)2 = E ( cp ) - E ( cp = 0 ) (in A unit) versus cp for F = 0.5. In 1 the point defects
energy is dominating. In 2 it is the energy saved by decreasing the curvature of the director field lines. In 3 the
anchoring effects dominate and the energy increases
again.
The wavelength is then proportional to the size of the capillary tube at a given temperature which has been verified experimentally. In this model, the transformation from a straight line into a zigzag has
a barrier of energy and is therefore thermically
activated. Note that this barrier is due to point
defects nucleation.
The height of the barrier needs to be calculated to know how easy is the transformation. Besides, an
exact calculation of B, C and cp * as functions of the elastic constants is necessary to predict the evolution
of the wavelength versus temperature. We are not able to perform such calculations but we can make
some predictions close to the transition.
3.4 ZIGZAG EVOLUTION NEAR THE SMECTIC-NEMA- TIC TRANSITION. - Firstly, it is useful to know the
evolution of the angle {3. As seen in section 3.2, j6
varies quickly in this range of temperature because of large variations of K12. More precisely, {3 is
constant and equal to -7fr very close to the smectic
phase (Tc T T + ) and decrease when T is greater than T+ :
Close to T+, E is a small angle and it is possible to develop B, the only constant which depends on /3, in
powers of E. B (f3
= 7r )
must vanish and the first 2non-zero term is :
Substituting B for its value in F leads to :
It is reasonable to assume that e varies faster than the ratio Fo in this range of temperature (near T+). From now on, we can consider Fo as a constant
and develop (AE)2 in powers of E. It yields :
Obviously, AE will have a negative minimum for
Fo 7/8.
Assuming this condition, the corresponding value Q * of Q is given by Q* = cp 2 E where cp 2 is the greatest solution of the equation :
The other solution Q 1 of this equation gives the
maximum of AE which scales like £ 2. Finally the wavelength is constant in our calculation and is given by :
1 77) k 1/3
In brief, this model shows qualitatively the exist-
ence of a narrow range of temperature (Tc-
T r+) in which the line is stable. Above T + the
zigzag shape develops spontaneously because the activation energy tends toward zero like 82. At the threshold (T = T+ ) its wavelength A * is finite and
independent of s whereas its amplitude (proportional
to cp A *) increases from zero proportionally to c.
This is in agreement with our observations near the smectic phase (see also Fig. 14).
3.5 CIRCULAR GEOMETRY : SPLITTING OF A S = 1
LINE NEAR A NEMATIC-SMECTIC TRANSITION. - For the compounds which have only a nematic phase (5CB, 6CB) we observe in the middle of the
