Zigzag disclinations in uniaxial nematic liquid crystals

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Zigzag disclinations in uniaxial nematic liquid crystals

Yves Galerne, Jacques Itoua, Lionel Liébert

To cite this version:

Yves Galerne, Jacques Itoua, Lionel Liébert. Zigzag disclinations in uniaxial nematic liquid crystals.

Journal de Physique, 1988, 49 (4), pp.681-687. �10.1051/jphys:01988004904068100�. �jpa-00210743�

Zigzag disclinations in uniaxial nematic liquid crystals

Yves Galerne, Jacques ^Itoua and Lionel Liébert

Laboratoire de Physique ^des Solides, ^Bât. 510, Université de Paris-Sud, ⁹¹⁴⁰⁵ Orsay Cedex, ^France (Requ le 23 octobre 1987, accepté le 17 dgcembre 1987)

Résumé.

Des lignes de disinclinaison en zigzags ^ont été observées dans des cristaux liquides nématiques

uniaxes (4-méthoxy-benzylidène-4’-n-butylaniline ^et 4-n-décyloxy ^{benzoate de} 4’n-hexyloxy phényl) qui

ressemblent beaucoup ^aux lignes zigzags déjà étudiées dans les nématiques biaxes. Elles sont cependant produites par ^un mécanisme différent. Il en résulte que les variations de l’angle ^des zigzags de disinclinaisons

en fonction de la température, ^sont différentes dans les nématiques ^{uniaxes et dans les} nématiques ^{biaxes. On}

en déduit un critère assez simple de biaxialité des phases nématiques.

Abstract.

Zigzagging disclination lines have been observed in uniaxial nematics (4-methoxybenzylidene-4’- n-butylaniline ^and n-hexyloxy-4’ phenyl n-decyloxy-4 benzoate), which very much resemble the zigzags already

studied in the biaxial nematics. They, ^however, originate ^{from a} different mechanism. It results a different temperature behaviour of the angle of the disclination zigzags in the uniaxial and the biaxial nematics, ^which

leads to a rather simple criterion for the biaxiality of the nematic phases.

Classification Physics ^Abstracts

61.30

61.70G

61.70J

1. Introduction.

Recently, 1 -disclinations have been observed in 2

biaxial nematic liquid crystals [1]. They ^{are lines} along which the local biaxial order is broken and reduced to the uniaxial order. The disclination lines in the biaxial nematics bear therefore a director, parallel ^{to them for} symmetry ^{reasons, which in-}

teracts with the anisotropic ^elastic field of the

surrounding liquid crystal. This interaction makes the disclination lines to orient along preferred ^direc-

tions and to keep ^the shape ^of zigzags. Because this mechanism needs two directors (one ^{to be broken on}

the line, and the other to realize the couplings ^{of the}

line with the bulk), ^{it has} reciprocally ^been suggested

that the zigzagging disclination lines are a character- istic feature of the biaxial nematics, and that their observation could indicate biaxiality in the nematic

liquid crystals [2].

In the paper ^we show that zigzagging disclination lines can also exist in the uniaxial nematics, ^{with a} slightly ^different mechanism. Contrarily ^{to what we}

believed earlier, their observation is therefore not an

absolute criterion of biaxiality [3]. ^{Other features,}

such as the temperature variations of the angle ^{of the} zigzags, ^{must be} considered as well to ascertain

biaxiality.

2. Experimental ^method.

The experimental part of this work is performed ^on thermotropic uniaxial nematics [4] : the well-known

4-methoxybenzylidene-4’-n-butylaniline (MBBA),

and n-hexyloxy-4’ phenyl n-decyloxy-4 ^benzoate (HPDB), ^{in the} temperature ranges : 22 °C to 47 °C and 80 °C to 86 °C, respectively. ^The experimental procedure ^{is the} following. ^{The nematic} compound

is inserted between two parallel glass plates ^treated

with silane in order to favor the homeotropic ^orien- tation, ^and separated ^with mylar spacers of known thickness. The sample ^is placed ^{inside a Mettler}

stage ^to regulate ^its temperature, and observed under a polarizing microscope. Applying ^a gentle

pressure ^on the cover slide of the sample ^{makes the}

nematic liquid crystal flow, and reorient in the shear flow thus created. This effect, ^{known for a} long time, results from the negative sign of the Leslie

viscosity coefficient _{a 3;} it exists in most of the available nematic liquid crystals, including ^MBBA [5]. ^{In this manner, it is} possible ^to put ^{a whole} domain of the sample ⁱⁿ planar orientation (as ^can

be tested by rotating ^the stage ^{of the} polarizing microscope) while the rest remains homeotropic (Fig. g ) 1). ^A disclination line of 1 -strength ₂ ^{is thus}

created at the boundaries between domains of

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01988004904068100

682

Fig. ^1.

^{Cut of a} homeotropic sample ^{submitted to local}

flow (arrow). ^A planar domain surrounded by ^{a discli-}

nation loop (hatched dots) appears. The lines sketch the orientations of the nematic director.

different orientations. (Disclination ^{lines of} higher

orders which appear ^as thicker lines under the

microscope, ^are also obtained in this _way, but we do not consider them here.) ^{In fact, the} planar ^orien-

tation is produced ⁱⁿ the bulk of the sample only. ^On

the glass ^surfaces, the nematic director is tilted because of the weak homeotropic anchoring. ^Be-

tween these two orientations in the vicinity ^{of the} glass surfaces, the orientation is distorted. Because the configuration ^of figure ^{1 is not} strongly ^anchored,

it slowly relaxes, making the disclination lines mi- grate, ^{and after a} long ^time disappear. ^{The whole} sample has then recovered its equilibrium ^{with the} complete homeotropic orientation. Before this oc-

curs, however, ^{we have the} opportunity ^to study ^the

behaviour of disclination lines under quasi-steady

conditions.

3. Transformation of the wedge ^to the twist discli- nation.

3.1 WEDGE-DISCLINATION LINE. - We restrict our

attention now to the symmetric ^{cases where the} 1 -disclination lines, ^which _separate ^{two domains of}

2

different orientations, ^are perpendicular ^{to the} plane

of both the planar ^and homeotropic orientations, and located at equal distances from the plates. ^{In the} simplest ^view, the molecules are arranged ^around

the so-called wedge-disclination ^{lines of} positive (Fig. 2) ^{or of} negative sign. (Both ^{cases are} equival-

ent within our approximations (see below) ; ^we

therefore explicitly consider the + 1 ₂ -wedge-discli-

nation lines only.) ^These configurations ^{do not}

involve the twist elastic deformation. Making ^the approximation ^{of twist} anisotropy only (Kl

K3

K =F K2) is therefore equivalent ^to making ^{the one-}

constant approximation ^{in these} specific ^{cases. The}

free energy per unit length ^{of the} wedge-disclination

line is therefore expressed [6] ^as

where D is the sample thickness, a is the diameter of the core of the wedge-disclination - ¹⁰⁰ A, Fc ^{is the}

Fig. ^2.

Wedge-disclination ^{line of} + 1-strength. ^To

simplify, the details of the anchoring ^{near the} plates ^are

not represented. ^The hatched dot schemes the core of the line.

energy per unit length ^{of this core, estimated} [6]

-

K/2, ^and FS is the surface energy per unit length

of the disclination line, ^{due to the} planar orientation in the left part ^{of the} sample. ^This Fs ^{term includes}

the anchoring energy, which results from the tilting

of the director onto the glass surfaces, ^{and the elastic}

energy of the distorsion in the vicinity ^{of the}

surfaces. Fs ^{results in a} transverse force which drives the line to the left boundary ^{of the} sample ^{until the} perfect homeotropic orientation is restored.

3.2 TWIST-DISCLINATION LINE. - In the uniaxial nematics, ^all the 1 ₂ -disclination lines can topologi- cally transform from one type ^{to another one, i.e.}

they ^all belong ^{to the same} homotopy ^class [7]. ^The wedge-disclination ^{described in} figure ^{2 can thus} continuously ^{transform into a} twist-disclination, ^at

least in its close vicinity ^{as sketched in} figure ^{3. In} fact, this transformation really ^{occurs if the free}

energy involved in the ^new orientation (Fig. 3) ^is

lower than the previous ^one (Fig. 2). This leads one to estimate FT, the free energy of the ^new orientation per unit length along ^{the z-axis, in order to} compare it to FW. FT is the addition of the surface _energy

Fs, ^{due to the} planar orientation as before, ^{to the}

bulk elastic _{energy of} the new distortion (Fig. 3).

This distortion is the superposition ^{of two} simpler

deformations : (I) ^a planar twist deformation which rotates the molecules parallel ^{to the} yz-plane ^around

the twist disclination line, ^and (II) ^a bend defor- mation in the left part ^{of the} sample ^{which rotates}

the molecules about the y-axis ^from parallel ^{to the} xy-plane ^to parallel ^{to the} yz-plane. ^The planar ^twist

deformation (I) ^{is a twist of wave vector} parallel ^to

the x-axis superposed ^{to a} splay, while the defor- mation (II) consists of a bend superposed ^{to a twist}

of wave vector parallel ^{to the} y-axis. Both defor- mations (I) ^and (II) ^{are therefore} orthogonal ^and

have additive elastic energies. ^{Let us now estimate}

these respective ^elastic energies ^{with the} assumption

Fig. ^3.

Twist-disclination line with the conventional

representation of the nematic directors as nails. The hatched parts ^{scheme the core} of the line. a) ^Cut along ^the xy-plane. b) ^Cut along ^the xz-plane. The dashed line represents ^the equilibrium direction of the line. The arrow

indicates the positive ^sense of orientation of the angles.

of twist anisotropy only Kl

K3

^{K =f::} K2. ^The

free _energy of the planar twist deformation (I) per unit length along the z-axis is given ^{from a} calculation

by Ranganath [8] :

The calculation of deformation (II) ^{has to be done} separately. The deformation (II) ^is specified by ^the angle ⁰ (x, y ) of the rotation around the y-axis ^{that it}

exerts on the molecules. e (x, y ) ^is thus solution of the Euler-Lagrange equation:

with the boundary conditions :

and 0 (x = 0, y = 0 ) = Ir - ^For the sake of simplici-

2

ty, it is useful to complement ^the boundary ^con-

ditions in the yz-plane. ^{We can thus assume that} 0 (x

^0, y) ^varies linearly between its two limits 0

and 7r , respectively at y

± D ^and y

0. The

2 p y 2

Fourier analysis ^{of this} problem ^{is then} simple. ^It

shows that, ^{for x 0 :}

the deformation (II) ^is therefore :

Limiting this series to its first term, ^we deduce the

expression of the free _energy per unit length ^{of the}

twist-disclination line to be :

where Fc, the energy per unit length ^{of the core of}

the twist-disclination, may be estimated - KK2/2.

3.3 MECHANICAL TRANSITION OF THE WEDGE TO THE TWIST-DISCLINATION. - As discussed above, the topological transformation of the + § or - § -

2 2

wedge-disclination ^{into a} twist-disclination _spon-

taneously ^occurs if the energy balance FT FW ^is satisfied, ^{i.e. for :}

This inequality shows that the wedge ^{to twist}

transformation appears for sample thicknesses D above some threshold value Do. Considering ^the published values of the elastic constants of MBBA

[9] and of other compounds [10] (which ^{show that}

K2 - _{2), Do} may be evaluated ^to be of the order of

K2

1 _)JLm. This makes the transformation easily ^observ-

able in the usual samples. ^{Let us} also notice that above the threshold of the external forces produced by ^the glass surfaces of the sample, ^the system ^loses

one symmetry element, ^{that is the} xy-mirror ^of symmetry (see for instance Fig.3b). ^{Such a} symmetry breaking ^{above threshold} typically indicates that the

wedge-twist transformation of disclinations is a

mechanical instability ^{or a} mechanical transition.

4. Tilting of the twist-disclination line.

The destruction of the xy-mirror ^of symmetry ^{at the} wedge-twist transition of the disclination (Sect. 3.3)

allows the disclination line to tilt by ^an angle 00 (dashed ^{line in} Fig. 3b), ^{and it} actually ^tilts

because of its elastic couplings ^{to the surfaces} through ^{the term} F2. ^{In order to} calculate the tilt

angle 00, let ^us estimate the _energy of the tilted disclination line per unit length ^of the z-axis. Keeping

the line located at the same position, the surface of the left part ^{of the} sample, ^and therefore the

anchoring energy Fs, remain the same as in sec-

684

tion 3. Consequently, Fs ^{does not} depend ^on 0 0. Because of the tilt, ^the length of the line

corresponding ^{to the unit} length ^{on the z-axis is} e2

multiplied by ^{a factor} of 1 ⁺ 2 ) ^as therefore the

B ² /

first and third terms of equation (6) ^{which become}

term of equation (6) ^{can be} calculated as in sec-

tion 3.2, just considering ^the amplitude change

(-f - ^{0 0 instead} of2 ) of the bend deformation (II)

due to the tilting ^{of the line. It becomes}

16 ( ^7T 00 ) ^{2 J- The} complete expression ^of 7T 2 00 ^{KK2. The complete} expression ^of

the twist-disclination energy per unit length ^{of the z-}

axis is therefore :

The tilt angle of the disclination line at equilibrium ^is

then deduced to be :

N - / N - I

Though the disclination line is tilted, if referred to the initial orientation of the sample, ^{it has to}

maintain its general direction, along the z-axis. The way for the line to satisfy ^this apparent paradox, ^is clearly ^{to make} zigzags ^{as in the} biaxial nematics [1].

The orientation of the director around the ^« _zag ^»

(the segment of line tilted by - 00) is deduced from the orientation around the « zig » (Fig. 3) by ^an xy- mirror of symmetry. This indicates that the corre-

sponding twist-disclinations have opposite signs ⁱⁿ

the « zigs » ^{and the «} zags ». Let ^us notice that the twist-disclinations discussed here originate ^{from the}

+ 1 ₂ -wedge-disclination ^of figure ^{2. Of course, simi-}

lar results are obtained when starting ^{from a}

1 -wedge-disclination, ₂ ^{g but with changed signs for}

the twist-disclinations of the «zigs» ^{and the}

« zags », compared ^{to their} signs ^{in the} + 1-wedge-

disclination case.

5. Experimental ^results.

5.1 GENERAL OBSERVATION. - Zigzagging ^discli-

nation lines have been obtained following ^{the exper-}

imental procedure ^{described above, in the whole} temperature range ^of the nematic phases ^{of MBBA}

and HPDM, ^{and for} sample thicknesses varying

from a few microns to 100 _f-Lm (Fig. 4). Except ^when hung ^{to a dust} grain, ^{the lines are} slowly moving ^due

to the relaxation of the planar anchoring. ^The

observation of this drift is important ^{to make sure}

that the lines ^are freely suspended ^{in the bulk, and}

not erratically ^{stuck on the} glass surfaces. The

velocity ^{of this} displacement is however _very slow

( ~ ¹⁰ f-Lm/s), ^{and does not} modify ^the equilibrium shape of the line as is exp,erimentally ^{verified when}

the line stops ^{for a while} (e.g. ^hindered by ^{some dust} aside). ^The samples ^are observed between crossed

polarizers ^oriented parallel ^to the sides of the

photographs. ^In figure 4a, ^the sample ^is placed ^so

that the planar region ^is extinguished, ^{which means}

that the planar direction is oriented perpendicular (or parallel) ^{to the} polarizers. ^In figures ^{4b and 4c,}

the sample ^{is tilted from this} original orientation. As

a result, ^the planar ^domain lightens ^{and a black} fringe appears along the disclination line. In all the cases, the general direction of the disclination lines,

or more exactly, ^the external bisectors of the zigzags,

are perpendicular ^{to the} planar direction revealed by

the polarizers, in accordance with the above dis- cussion (end ^{of Sect.} 4).

5.2 BLACK FRINGES.

As denoted above, ^the disclination line _appears as decorated with black

fringes ^{when the} sample ^is slightly rotated from its

extinguished position (between ^crossed polarizers).

More specifically, ^the fringes follow the ^« zigs » ^{for a} positive ^rotation (photo 4b), and follow the ^« zags » for a negative rotation of the sample (photo 4c).

This is an optical effect due to birefringence. ^The fringes (which ⁱⁿ principle ^are grey and not black)

reveal the places ^{where the} average direction of the molecules is oriented along ^the polarizers. ^The

measurements of the abscissa x’(= - x ) ^{of the} fringe

referred to the disclination line as a function of the

angle 0 of rotation of the sample ^are given ⁱⁿ figure ⁵

for a sample of thickness D

45 tLm. 0 (x’ ) appears thus to vanish exponentially ^{with a} characteristic

length 6 - ^D.

This result is to be compared with the average value of 0 over the sample thickness. Restricting ^the

calculations to first order, ^{we find from} equation (4) :

This expression is rather consistent with the exper- imental results of figure 5, ^{if one takes into account}

that : 2 - ; . It therefore yields ^{a direct} support ^to

7T 4

the model of the bend distortion in the planar region, with the molecules arriving tangent ^{to the}

twist-disclination line (Fig. 3b). However, ^one may notice that the experimental ^{and the calculated}

characteristic lengths ^differ by ^{a factor} of about 2.

Fig. 5.

^Rotation angle ^{ø of} the sample ^{versus the}

abscissa x’(= - x) of the black fringe, ^{in a} sample ^of

thickness D

45 _Rm.

This significant departure ^from the model could indicate that the planar orientation of the sample ^is

not so strongly ^{held at the} glass ^{surfaces as} supposed

in section 3.

5.3 ZIGZAG ANGLES.

The angles ^{of the} zigzagging

disclination lines present large variations from one to another. Such variations could originate ^from

hazardous surface perturbations unsufficiently ^con-

trolled by ^{the too weak} planar alignment ^{of our} samples (as already suspected ^{above to} explain ^the

increased values of g). ^These variations being ^erratic,

it is possible ^to filter them statistically. ^{We thus used}

video recordings ^{to make} data accumulations (up ^to

30-50 in each case). ^These experimental ^{means are}

also _very convenient to select zigzags ^at equilibrium

and to take series of pictures ^{of the same} zigzag ^for

different rotation angles ^{of the} microscope stage ^to

measure the black fringe (Sect. 5.2).

The zigzag angles have been measured far from the peaks ^to avoid the distortions around it [11], ^and

under different conditions of temperature ^and

sample ^{thickness for both the MBBA and the HPDB} compounds (we ^{used then a cleaner source than for}

Fig. ^4.

Photomicrographs ^of zigzagging disclination lines between crossed polarizers, ^oriented horizontally ^and vertically. ^The homeotropic ^and planar parts ^{of the} samples ^are respectively ^{on the} right ^{and on} the left sides of the photos. ^The photos ^are taken with the same

magnification. The distance of the black fringe ^{to the}

disclination line is typically of the order of the sample

thickness. a) ¹⁰ lim-thick sample ^{of HPDM at 85} °C,

oriented perpendicular ^{to the} polarizer. b) ⁴⁵ JLm-thick sample ^{of MBBA at 35} °C, ^rotated by ^{a small} positive angle (following ^{the sense of the arrow in} Fig. 3b) ^referred

to the direction of the polarizer. c) ⁹⁰ tlm-thick sample ^of

HPDM at 81 °C rotated by ^{a small} negative angle.

686

the photos). ^No dependence ^{of the} zigzag angles ^on

the temperature, ^{nor on the} compound ^{has been}

observed. This observation is clearly consistent with

equation (9). ^{However, one} may notice that our

calculations are based on the approximation Kl

K3, ^which experimentally ^is rarely ^verified (e.g. ⁱⁿ

MBBA [9], K3/ Kl -- 1.5). ^{An exact} calculation, releasing ^this approximation, ^would express the tilt

angle 0 0 ^{of the} disclination line as a function of the ratios of the three elastic ^constants Kl, K2 ^and K3. Through them, 00 ^{would in fact} slightly depend

on the nature of the compounds, ^{and on the} temperature. These variations should however re-

main negligible ^if compared ^{to 00 itself.}

The only variation of the zigzag angle 00 ^{that we}

have found is a regular increase with the sample

thickness D. Figure ^{6 shows the} measurements of

1/00 ^{versus In} (Dla) ^{for MBBA} (corresponding ^to

D ranging ^{between 5 to 100} Rm), together ^{with the}

calculated variations from equation (9) (straight line). ^The agreement ^{between the} experimental ^and

the calculated variations is correct within the stan-

dard deviations of the data (represented ^as bars).

The curvature in the variations of the experimental

data that one could notice in figure ^{6 is} probably ^not significant. ^It could however be due to higher ^order

terms that we have neglected in the calculations (e.g.

in Eqs. (1) ^and (2)).

Fig. 6. - Inverse tilt angle 1/80 of the disclination line

versus In (D/a ). ^The experimental ^{data are} represented

with their standard deviations (500/00-3-4%, ^and

BD - few microns). ^The straight ^line corresponds ^{to the}

calculated variations from equation (9).

6. Discussions.

The zigzagging disclination lines, reported ^{here in}

the uniaxial nematics, resemble those already ^ob-

served in the biaxial ones [1]. ^We may however try

to find differences which could be used as criteria for the biaxiality. ^A difference that one can first notice,

is the black fringe discussed here in details, ^and unobserved in the biaxial nematic phase [1]. ^{In fact,}

as discussed in section 5.2, this black fringe ^is

relevant to the value of g, ^{which is} determined by ^the experiment, ^{and not to} the uniaxial or biaxial nature of the liquid crystal. In reference [1], ^the sample ^was

oriented with a magnetic ^{field H which} imposed ^a small g. ^{As a} result, ^{the black} fringe ^{was stuck to the}

disclination line, ^{and thus not} observable. The observation (or non-observation) of the black fringe along ^a zigzagging disclination line is therefore not related to uniaxiality (or biaxiality).

A difference between uniaxial and biaxial nematics is the temperature variations of the angles ^{of the}

disclination zigzags. ^{In the biaxial nematics, 00} strongly depends ^on temperature, because it is

directly proportional ^to the biaxial order parameter [1]. ^In the uniaxial nematics as discussed in section 5.3, 00 ^is independent ^{of, or} slightly dependent ^on

temperature. ^This suggests ^{to use the} temperature

variations of the zigzags ^{as an} indication of biaxiality [3]. ^Such measurements could be an easier task than

succeeding ^an undoubtless conoscopic figure [12]

(which ^{needs a} perfectly ^oriented sample).

As discussed in section 5.3, ^we suspect ^{that the}

large uncertainties of our measurements arise from the too weak planar alignment ^{of our} samples. ^It

would thus be useful to improve ^the planar ^orien-

tation by applying ^a horizontal magnetic ^{field on the} sample. ^The expression ^of 00 would then be modified

by essentially replacing D/2 ⁱⁿ equation 9 b ^the magnetic correlation length g H

K _XaH ^[6]

(which ^{becomes the} pertinent length ^{of the} prob- lem).

Applying ^a magnetic ^{field onto the} sample ^would

also be interesting ^to observe the continuous trans- formation of the disclination lines. When the magne- tic field is increased up ^{to a} few kiloGauss, ç H is decreased to a few microns (since KIYA -

5 dynes), pushing the black fringe ^{close to the}

disclination line. The black fringe would then disap-

pear, making ^the zigzag resemble very much the ^one observed in the biaxial nematics. The decrease of

6H would also produce ^{an increase of the} zigzag angle 00 ^{until the} mechanical transformation of the twist to the wedge-disclination ^{line occurs} (Sect. 3.3). ^{At the threshold} value H - 150 kG

(from Eq. (7), replacing D/2 by gH)’ ^{the line} straightens, ^i.e. 00 jumps ^from its maximum value

( ^{= 23° from} Eq. (9)) ^{to zero. The twist to} wedge-

disclination transformation is therefore a first-order mechanical transition. It is worth noticing ^{that such}

an experiment ^could probably ^{be more} easily per-

formed using lyotropic ^nematic liquid crystals doped

[13].

In conclusion, ^we have observed disclination lines in the form of zigzags ^{in the uniaxial} nematics, ^which resemble very much the zigzags already observed in the biaxial nematics. The zigzagging disclinations in the uniaxial nematics, however, ^are produced by ^a

different mechanism than in the biaxial nematics.

They result from the elastic interaction between two

conflictual orientations of the director far from ^the

disclination line and in its immediate vicinity, ^{and not} strictly ^{on it as in the biaxial nematics.} (This ^is impossible ^{for the} disclinations in the uniaxial nema-

tics because they ^{have an} isotropic ^{core, and thus}

director, ^i.e. typically in the uniaxial nematics. Let

us notice that other conflictual configurations ^than

that studied here, ^could similarly produce zigzagging

disclination lines in the uniaxial nematics. For in- stance, ^a wedge-disclination (provided that it is

properly stabilized) ^{could also make} zigzags ^if pro- duced in a planar sample ^{and oriented} parallel ^{to this} planar alignment.

It is a pleasure ^{to thank Drs. G.} Durand, ^Ph.

Martinot-Lagarde ^{and P.} Pieranski for _very helpful discussions, ^{and Dr. C.} Germain for providing ^us

with MBBA and HPDB.

References

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[2] MALTHÊTE, J., LIÉBERT, L., LEVELUT, ^{A. M. and} GALERNE, Y., C. R. Acad. Sci. 303 (1986) ^1073.

[3] GALERNE, Y., Colloque d’expression française ^sur

les Cristaux Liquides (Arcachon), septembre

1987 (to ^be published).

[4] Similar studies in lyotropic uniaxial nematics are

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[5] GÄHWILLER, C., Phys. Rev. Lett. 28 (1972) ^1554.

[6] ^{DE GENNES, P. G., The} Physics of Liquid Crystals (Clarendon, Oxford) ^1974.

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[8] RANGANATH, G. S., Mol. Cryst. Liq. Cryst. ⁸⁷ (1982) ^187.

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A. M. J., ^Mol. Cryst. Liq. Cryst. ³⁷ (1976) ^269.

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Phys. ⁶⁷ (1977) ^3705.

SCHAD, ^{H. and} OSMAN, ^M. A., ^{J. Chem.} Phys. ⁷⁵ (1981) ^880.

[11] GALERNE, Y., ^{to be} published.

[12] GALERNE, Y., ISAERT, N., LEVELUT, A. M., ^LIÉ-

BERT, L. and MALTHÊTE, J., Colloque d’expres-

sion française ^{sur les Cristaux} Liquides (Arca- chon), septembre ¹⁹⁸⁷ (to ^be published).

[13] ^{KROIN, T.} and FIGUEIREDO NETO, ^A. M., Phys.

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