Cusp shaped hydrodynamic instability in a nematic

(1)

HAL Id: jpa-00209493

https://hal.archives-ouvertes.fr/jpa-00209493

Submitted on 1 Jan 1982

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Cusp shaped hydrodynamic instability in a nematic

E. Guazzelli, E. Guyon

To cite this version:

E. Guazzelli, E. Guyon. Cusp shaped hydrodynamic instability in a nematic. Journal de Physique,

1982, 43 (7), pp.985-989. �10.1051/jphys:01982004307098500�. �jpa-00209493�

(2)

985 Cusp shaped hydrodynamic instability ⁱⁿ ^a ^nematic

E. Guazzelli (*) ^and ^E. Guyon

Laboratoire d’Hydrodynamique ^et Mécanique Physique (**), E.S.P.C.I., 10, ^rue Vauquelin, ⁷⁵²³¹ Paris, ^France (Reçu le 24 décembre 1981, accepté ^{le 9} ^mars 1982)

Résumé.

²⁰¹⁴

Nous étudions les seuils de l’instabilité hydrodynamique ^d’un nématique homéotrope ^{soumis à} ^un

cisaillement elliptique. ^Nous analysons ^le ^rôle ^du changement ^de symétrie causé par la variation de l’ellipticité

du cisaillement au voisinage ^du ^cas dégénéré ^{de la} polarisation circulaire.

Abstract.

²⁰¹⁴

We study ^the hydrodynamic instability thresholds of a homeotropic ^nematic subjected ^to ^an elliptical

shear. We analyse the role of the changes ^{in the} symmetry ^caused by the variation of the ellipticity of the shear around the degenerate ^circular polarization.

LE JOURNAL ^DE PHYSIQUE

J. Physique ⁴³ (1982) ^985-989 ^JUILLET 1982,

Classification

Physics Abstracts

61.30E - 47.20

1. Introduction.

^-

The effect of an elliptical ^shear

on a homeotropic layer ôf ^nematic material has been the subject ôf â variety ôf experimental studies : des-

cription ôf ône dimensional (rolls) [1, 2] ând ^two ^dimen-

sional (squares, hexagons) [3] instability patterns;

« elasticity >> of the roll pattern ^{from the} study ^of

individual defects of the structure [4] ; ^« melting » ^of

the two dimensional structures [5]. ^{In this} article, ^we discuss the effect of the changes ⁱⁿ ^the symmetry ^caused

by variation of the ellipticity ^{of the} ^shear, ^which plays

the role of an external field for the direction of the convective rolls.

2. Experimental techniques.

^-

^The elliptical ^shear

is produced by applying ^a ^linear alternating ^shear ^to

the lower (L) and upper (U) plates limiting the cell of thickness d (Fig. 1) :

The nematic is held by capillary between the horizontal

plates ^which âre âttached ^{to two} loudspeakers ând

which can be moved vertically by ^a micrometer.

The control of parallelism and of the thickness d is made interferometrically.

(*) ^Also ONERA, 29, ^avenue de la Division Leclerc,

92320 Chatillon, France.

(**) ^ERA ^no ^1000.

Fig. ^1.

^-

Experimental apparatus : ^a ^nematic liquid crystal

is sandwiched between two rectangular plates. ^The ^thickness of the cell is d (= ³⁰ ^to ¹⁵⁰ gm). ^The ^two plates ^move ⁱⁿ quadrature ^at ^the frequency f ( = ⁹⁰ ^to ⁵⁰⁰ Hz).

The frequency f is defined from two oscillators whose output signals excite the loudspeakers. ^The homeotropic alignment (n perpendicular ^to ^the plates

in the absence of shear) îs înduced by treating ^the înner

faces of the flow cell with lecithin.

The parameter controlling ^the instability ^{is :}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01982004307098500

(3)

986 where D is a diffusivity for the nematic orientation.

N has the structure of a Reynolds number for the

problem.

The liquid crystal used is MBBA [6] (p. methoxy-

benziliden _p.n. butylanilin) which is nematic from 18 to 46 OC. Problems of temperature stability arise from the rapid ^variation of elastic (K ) [7] and viscous (y) [8]

coefficients with temperature. ^For â ^variation ôf ^T ôf

1 OC, ^we êstimate â ^variation ôf ^D ⁼ K/y

^-

^and

of the threshold

^-

of 3 % ^that ^we have verified expe-

rimentally. Ôur measurements were done at 22 °C with a temperature accuracy better than 1 OC by circulating âir regulated ⁱⁿ temperature ⁱⁿ â ^closed volume surrounding the flow cell and the microscope through which observations âre made. In addition,

the poor chemical stability ^{of this} compound ^{causes a}

drift of properties ôver â period ôf ^several days;

consequently, ^data ^are ^taken ^{over a} restricted period

of time. The experiments ^are performed ^for increasing

values of N keeping ^a ^constant ellipticity ^E ⁼ Xo/Yo.

The value of the thresholds are determined by ^observ- ing ^the light diffracted by ^the grating ^formed by ^the

convective pattern using ^a laser incident at right angle ^on ^{the cell.}

3. Instability thresholds.

^-

We determine two thres- holds : N 1, the lowest-linear-one above which rolls ^are obtained continuously ^{and in} ^a reversible _way;

N2 ( > N1 ) ^where â ^two dimensional pattern, ^consist- ing usually ôf squares caused by secondary ^rolls ât right angles the initial _ones, appears reversibly.

An instability plot in coordinates (Xo, Yo) (Fig. 2)

shows deviation for small d or f ^{from the} hyperbolic

Fig. ^2.

^-

Instability diagram : variation of the roll (full

dots (b) and squares (a)) ^and ^square pattern (open ^dots (b)

and squares (a)) ^threshold in coordinate (Xo, Yo) :

shape predicted by (2) : ^a cusp is observed for the roll threshold near the 45° line corresponding ^to ^the

value E ⁼ 1 where the shear becomes circular in the bulk of the cell.

The existence of two branches symmetrical ^with respect ^to this axis is obvious from symmetry ^conside- rations as is also substantiated by the observation of the direction of the rolls (Fig. 3) : in the lower octant

(E 1 ), the rolls form an angle g/ - 20° with the x axis nearly independent ^of E; ^the upper ^octant (E > 1)

is obtained by exchanging the role of x and y ^{in the} experiment, ^the angle ^is

Fig. ^3.

^-

Variation of the direction of the rolls as a function of the ellipticity of the shear.

This experimental ^value observed also by [1, 3] ^{is much} larger than the theoretical one 4/ - ³⁰ (J. ^M. Dreyfus

and C. D. Mitescu [9] ^have introduced in the calcula- tion [2] convective terms which _may explain ^this disagreement). ^The existence of a quasi discontinuous

change ^of angle 4/ (Fig. 3) around the value E ⁼ 1

(circular polarization) ^is associated with the change ^of

symmetry ^{of the} hydrodynamic shear around this

particular ^value. ^It ^is ^thus reasonable to relate the cusp in the threshold ^curve ^to the effect of the dege-

neracy around the E ⁼ 1 value.

J. Sadik and coworkers [10] ^have recently proposed

an analysis ^{of this} cusp. We will not reproduce ^their argument ^which ^is ^based ^{on an} ^extensive development

of the above symmetry argument (in particular

x --+ y and 0 --+ 7r/2 - 4/) applied ^to the linear solu-

(4)

tion of the nematodynamic problem. We will write instead an « ad hoc » equation ^for ^threshold ^which reproduces their main result. The analysis leading

to the form (2) ^assumes that the ratios Xo/d, Yo/d,

which represent ^the normalized displacements ^{of the} plates ^are ^small. ^It is also the limit where the alternat-

ing ^director distortions are proportional ^to ^{the shear.}

For larger ^shear amplitudes, ^non ^linear contributions

already appear in the alternating distortion below threshold [2]. ^{In such} conditions, ^the ^threshold ^can ^be

written as a symmetric expansion ^of (2) in powers of

X o/d ^and Yold (formula ⁽⁵⁵⁾ ^of [10]) :

The absence of linear term comes from the invariance of the problem ^with changes ^{x - -} ^x, y - - y. The absolute value in the right hand side of (3) implies

that two solutions are met depending ^on ^whether Xo Z Yo. A ^{and B} ^are functions of the material

properties. În principle, they ^can ^be ôbtained ^{from the} complex ^linear analysis. Â determines the amplitude

of the cusp, B that of the _cusp angle ² ^0. ^*

From the study ^{of the} ^two branches of solution of (3) along ^{the line} Xo = Yo, ^one gets ^an angle (Fig. 2)

between the tangents given ^for large d and f, by (formula (57) ^of [10]) :

with

We have analysed ^the validity of the dimensional form (4) by studying the threshold curves for nematic films of different thicknesses (from ^d ⁼ ³⁰ ^to ¹¹⁰ pm)

at different frequencies (from f ⁼ ⁹⁰ ^to ⁵⁰⁰ Hz). ^The

results concerning ^the cusp ^are compared ^{with the} analysis leading ^to ^formula (4) ^on ^three figures.

Figure 4, ôbtained ^for â ^constant value of d ⁼ 50 _gm shows that the _cusp angle ^decreases âs f -1 ^when f

increases. On figure 5, ^for ^a ^constant frequency

Fig. ^4.

^-

Variation of the cusp angle ^with frequency ^at

fixed thickness. The slope of the broken line is

^-

1. Fig. ^5.

^-

Variation of the cusp angle with the thickness at fixed frequency. ^The slope of the broken line is

^-

2. f ⁼ ¹⁵⁰ Hz, ^we check that the cusp angle ^decreases

as d - 2 when the thickness d increases. For large d,

the variation of the threshold barely departs ^from ân hyperbolic ône (as in the form (2)) ^{and the} ^cusp îs practically âbsent. ^For ^d ⁼ ¹¹² Ilm, ^we find

in agreement ^{with the} values measured in [1, 3]. Large

values of the _cusp angle âre ôbtained for small film thicknesses. The limit of validity ^{of the} analysis ôccurs

when the thickness becomes smaller than a value of - the order of the amplitudes ^of displacement ^{of the} plates. ^On figure 6, ^we ^have summarized the variation of the cusp angle for different f and d, which agrees with the expression (4) in the limit of validity (large f

and d) ôf the calculation of [10]. ^The slope ^K ^{of the} straight ^line ôn figure ⁵ gives â ^{value B} ⁼ ¹² ⁺ ^5.

In order to analyse ^the step (Fig. 2) ^between ^two

different threshold curves (a) ^and (b), ^{we can} ^write

the reduced difference [11] ^for ^E ⁼ ¹ according ^to (3) :

with

The experimental plots ⁱⁿ figure ⁷ ^show ^a large

scatter of results which can be fitted to a linear law

giving ^a value A - 3 defined with a factor of 2 accu-

racy. This is ^not surprising ^as ^different results obtained at different times _may have a superimposed ^{shift of}

threshold due to the lack of stability ^{of the} nematic;

this gives ân amplified êffect ôn ^the ^curve expressing

a difference between two thresholds.

4. Discussion.

^-

The theoretical model [10] ^based

on symmetry considerations using linearized _equa-

tions of mean field type predicts : ^an influence of the

(5)

988 Fig. ^6.

^-

Variation of the cusp angle ^{with the} frequency /

and thickness d. The dotted line is the best fit to determine the slope ^K.

ellipticity ^on the orientation of the roll structure, ^the existence of a cusp effect and its dependence ^{with d}

and f ⁱⁿ agreement ^{with the} experiments.

From this linear analysis ^{it is} ^not possible ^to înfer ân important result visible on figure ^{2 : due} ^to ^the cusp ôn the threshold curve N 1 ând ^to the much smaller (or

non existing) ^one ^on ^the ^threshold N2 ^for ^two ^dimen-

sional structures, the solution N1 ^{is close} ^to ^the ^two

dimensional one for circular shears. The difference between the two thresholds increases when the fre- quency ^or the thickness decreases. When the shear is increased for this state of polarization (E ⁼ 1), ^the

formation of squares follows very rapidly ^that ^of

rolls as if the later structure was only ^a ^transient ^one.

Linear analysis only ^describes patterns ^of parallel

rolls. Non linear terms characterize interactions between rolls of different directions and determine in particular ^the conditions of formation of two

dimensional structures. Due to the anisotropy ^intro-

duced by ^the elliptic shear, ^a preferred direction of rolls is obtained which is predicted by the linear

analysis. ^{It is} reasonable to assume that, ^{in the} ^case E # 1, the solution is separated from those implying superposition ôf ^rolls ât ^different angles by â ^finite

amount. On the other hand, linear solutions of diffe- rent directions âre equally probable ^{in the} ^case ôf â degenerate circular shear. This can explain why ^the

square structure is favoured for E ⁼ 1.

Fig. ^7.

^-

Variation of the left hand of equation (6) ^{as a}

function of [Xg(a)/d2(a) - Xg(b)/d2(b)]. ^The ^{index b} corresponds ^{to d} ⁼ ^{112 ±} 5 -gm, f ⁼ 150 Hz. The index a

corresponds ^{to d = 30} ^to 112 urn ^at f ⁼ ⁹⁰ ^to ^{500 Hz.}

The dotted line gives ^an ^{order of} magnitude ^{of the} slope ^P.

T he result is reminiscent of the problem ^of Rayleigh-

Benard instability ⁱⁿ ^a homeotropic nematic heated from above [12] recently analysed by ^M. Gabay [13].

In this problem, the solution just above the critical temperature difference consists of crossed rolls. If the

degeneracy ^{of the} problem ^is suppressed by applying

a small horizontal magnetic ^field ^H which tends to

align ^molecules along it, the solution at threshold consists of rolls perpendicular ^to the direction of the field H. However, ^no appreciable difference has been detected between the value of the roll threshold as

extrapolated ^{from the} threshold ATC(H) ^to ^H ⁼ ⁰

and that of the _squares AT,, ^(H = 0). ^{In both} pro- blems in the degenerate ^case (E ⁼ 1, ^H ⁼ 0), ^two symmetries ^are ^broken ^at ^the ^same ^{time due} ^to ^the

formation of rolls : one giving the direction of rolls and the usual one corresponding ^to ^the adjustment

-

the phase

^-

^{of rolls} ôf â given ^wave ^vector. ^The coupling ôf ^the ^two effects leads to a new term

^-

the

so called Brazovskii field [14]

^-

^{in the} ^non ^linear problem, ^which ^is expected theoretically ^to ^force

a first order transition (inverse bifurcation) ^{instead of}

the second order one (direct bifurcation). În êither problem (E ⁼ 1, ^H ⁼ 0), ône ^has ^not ^{been able} ^to

detect _any irreversible behaviour in the threshold

curve within the _accuracy of the experiments (- ¹ %).

(6)

However we have noted that, ^{in the} present experi-

ments, ^the square pattern often appears ^to be unstable

to hexagons ^not ^{far above} N2 ^but ^not ⁱⁿ ^a ^uniform

fashion. This is particularly important ^for ^E ⁼ ^1.

This suggests ^a first order transition as also expected

from symmetry considerations [15].

Acknowledgments.

^-

^We ^are ^indebted ^to ^J. ^Sadik

for a helpful correspondence ^on ^the subject ^{of this}

article and to E. Dubois-Violette, ^M. Gabay, ^and

F. Rothen for many useful discussions. The collabora- tion of. I. Janossy, ^the ^{skill of} ^M. Clement and J. M. Dreyfus ⁱⁿ ^the experiment ^are ^also acknowledged.

References

[1] PIERANSKI, P., GUYON, E., Phys. Rev. Lett. 39 (1977)

1281.

[2] DUBOIS-VIOLETTE, E., ROTHEN, F., ^J. Phys. ¹⁰ (1978)

1039.

[3] DREYFUS, ^J. M., GUYON, E., J. Physique ⁴² (1981) ^459.

[4] GUAZZELLI, E., GUYON, E., ^WESFREID, ^{J. E. in} ^« Sym- metry and Broken Symmetry ⁱⁿ ^Condensed ^Matter Physics ^», Ed. N. Boccara (IDSET Paris) ^1981.

[5] GUAZZELLI, E., GUYON, E., C.R. Hebd. Séan. Acad.

Sci. 292 (1981) ^142.

[6] ^{The MBBA} used in the experiments ^{has been} synthe-

sized by L. Strelecki.

[7] ^DE GENNES, ^P. G., ^The Physics of Liquid Crystals (Cla-

rendon Press, Oxford) 1974, p. 66.

[8] GÄHWILLER, ^C. H., ^Mol. Cryst. Liq. Cryst. ²⁰ (1973)

301. [9] DREYFUS, J. M., thèse de 3e cycle, ^{Paris VI} (juin 1979) ; Chap. 5 in collaboration with C. D. MITESCU.

The difference between the case where the shear is

produced by applying ^two linear shear at right angle ^to ^the plates (form (1)) ^and ^the ^one [2], [10]

where the full motion is applied ^to ^one plate,

is characterized by the presence of different convec-

tive terms.

[10] SADIK, J., ROTHEN, F., BESTGEN, W., DUBOIS-VIOLETTE, E., ^J. Physique ⁴² (1981) 915. The convective terms have been omitted in the calculation.

[11] SADIK, J., ^Private communication.

[12] SALAN, J., Thèse, ^Madrid (1982).

[13] GABAY, M., Thèse, Orsay (juin 1981).

[14] BRASOVSKII, ^S. A., Sov. Phys. ^{JETP 41} (1975) ^85.

[15] BUSSE, ^F. H., Rep. Prog. Phys. ³⁰ (1978) ^629.

Cusp shaped hydrodynamic instability in a nematic

HAL Id: jpa-00209493

https://hal.archives-ouvertes.fr/jpa-00209493

Submitted on 1 Jan 1982

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Cusp shaped hydrodynamic instability in a nematic

E. Guazzelli, E. Guyon

To cite this version:

E. Guazzelli, E. Guyon. Cusp shaped hydrodynamic instability in a nematic. Journal de Physique,

1982, 43 (7), pp.985-989. �10.1051/jphys:01982004307098500�. �jpa-00209493�

985

Cusp shaped hydrodynamic instability in a nematic

E. Guazzelli (*) and E. Guyon

Laboratoire d’Hydrodynamique et Mécanique Physique (**), E.S.P.C.I., 10, rue Vauquelin, 75231 Paris, France (Reçu le 24 décembre 1981, accepté le 9 mars 1982)

Résumé.

Nous étudions les seuils de l’instabilité hydrodynamique d’un nématique homéotrope soumis à un

cisaillement elliptique. Nous analysons le rôle du changement de symétrie causé par la variation de l’ellipticité

du cisaillement au voisinage du cas dégénéré de la polarisation circulaire.

Abstract.

We study the hydrodynamic instability thresholds of a homeotropic nematic subjected to an elliptical

shear. We analyse the role of the changes in the symmetry caused by the variation of the ellipticity of the shear around the degenerate circular polarization.

LE JOURNAL DE PHYSIQUE

J. Physique 43 (1982) 985-989 JUILLET 1982,

Classification

Physics Abstracts

61.30E - 47.20

1. Introduction.

The effect of an elliptical shear

on a homeotropic layer of nematic material has been the subject of a variety of experimental studies : des-

cription of one dimensional (rolls) [1, 2] and two dimen-

sional (squares, hexagons) [3] instability patterns;

« elasticity &#x3E;&#x3E; of the roll pattern from the study of

individual defects of the structure [4] ; « melting » of

the two dimensional structures [5]. In this article, we discuss the effect of the changes in the symmetry caused

by variation of the ellipticity of the shear, which plays

the role of an external field for the direction of the convective rolls.

2. Experimental techniques.

The elliptical shear

is produced by applying a linear alternating shear to

the lower (L) and upper (U) plates limiting the cell of thickness d (Fig. 1) :

The nematic is held by capillary between the horizontal

plates which are attached to two loudspeakers and

which can be moved vertically by a micrometer.

The control of parallelism and of the thickness d is made interferometrically.

(*) Also ONERA, 29, avenue de la Division Leclerc,

92320 Chatillon, France.

(**) ERA no 1000.

Fig. 1.

Experimental apparatus : a nematic liquid crystal

is sandwiched between two rectangular plates. The thickness of the cell is d (= 30 to 150 gm). The two plates move in quadrature at the frequency f ( = 90 to 500 Hz).

The frequency f is defined from two oscillators whose output signals excite the loudspeakers. The homeotropic alignment (n perpendicular to the plates

in the absence of shear) is induced by treating the inner

faces of the flow cell with lecithin.

The parameter controlling the instability is :

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01982004307098500

986

where D is a diffusivity for the nematic orientation.

N has the structure of a Reynolds number for the

problem.

The liquid crystal used is MBBA [6] (p. methoxy-

benziliden p.n. butylanilin) which is nematic from 18 to 46 OC. Problems of temperature stability arise from the rapid variation of elastic (K ) [7] and viscous (y) [8]

coefficients with temperature. For a variation of T of

1 OC, we estimate a variation of D = K/y

and

of the threshold

of 3 % that we have verified expe-

rimentally. Our measurements were done at 22 °C with a temperature accuracy better than 1 OC by circulating air regulated in temperature in a closed volume surrounding the flow cell and the microscope through which observations are made. In addition,

the poor chemical stability of this compound causes a

drift of properties over a period of several days;

consequently, data are taken over a restricted period

of time. The experiments are performed for increasing

values of N keeping a constant ellipticity E = Xo/Yo.

The value of the thresholds are determined by observ- ing the light diffracted by the grating formed by the

convective pattern using a laser incident at right angle on the cell.

3. Instability thresholds.

We determine two thres- holds : N 1, the lowest-linear-one above which rolls are obtained continuously and in a reversible way;

N2 ( &#x3E; N1 ) where a two dimensional pattern, consist- ing usually of squares caused by secondary rolls at right angles the initial ones, appears reversibly.

An instability plot in coordinates (Xo, Yo) (Fig. 2)

shows deviation for small d or f from the hyperbolic

Cusp shaped hydrodynamic instability ⁱⁿ ^a ^nematic

E. Guazzelli (*) ^and ^E. Guyon

Laboratoire d’Hydrodynamique ^et Mécanique Physique (**), E.S.P.C.I., 10, ^rue Vauquelin, ⁷⁵²³¹ Paris, ^France (Reçu le 24 décembre 1981, accepté ^{le 9} ^mars 1982)

Nous étudions les seuils de l’instabilité hydrodynamique ^d’un nématique homéotrope ^{soumis à} ^un

cisaillement elliptique. ^Nous analysons ^le ^rôle ^du changement ^de symétrie causé par la variation de l’ellipticité

du cisaillement au voisinage ^du ^cas dégénéré ^{de la} polarisation circulaire.

We study ^the hydrodynamic instability thresholds of a homeotropic ^nematic subjected ^to ^an elliptical

shear. We analyse the role of the changes ^{in the} symmetry ^caused by the variation of the ellipticity of the shear around the degenerate ^circular polarization.

LE JOURNAL ^DE PHYSIQUE

J. Physique ⁴³ (1982) ^985-989 ^JUILLET 1982,

The effect of an elliptical ^shear

on a homeotropic layer ôf ^nematic material has been the subject ôf â variety ôf experimental studies : des-

cription ôf ône dimensional (rolls) [1, 2] ând ^two ^dimen-

« elasticity >> of the roll pattern ^{from the} study ^of

individual defects of the structure [4] ; ^« melting » ^of

the two dimensional structures [5]. ^{In this} article, ^we discuss the effect of the changes ⁱⁿ ^the symmetry ^caused

by variation of the ellipticity ^{of the} ^shear, ^which plays

^The elliptical ^shear

is produced by applying ^a ^linear alternating ^shear ^to

plates ^which âre âttached ^{to two} loudspeakers ând

which can be moved vertically by ^a micrometer.

(*) ^Also ONERA, 29, ^avenue de la Division Leclerc,

(**) ^ERA ^no ^1000.

Fig. ^1.

Experimental apparatus : ^a ^nematic liquid crystal

is sandwiched between two rectangular plates. ^The ^thickness of the cell is d (= ³⁰ ^to ¹⁵⁰ gm). ^The ^two plates ^move ⁱⁿ quadrature ^at ^the frequency f ( = ⁹⁰ ^to ⁵⁰⁰ Hz).

The frequency f is defined from two oscillators whose output signals excite the loudspeakers. ^The homeotropic alignment (n perpendicular ^to ^the plates

in the absence of shear) îs înduced by treating ^the înner

The parameter controlling ^the instability ^{is :}

benziliden _p.n. butylanilin) which is nematic from 18 to 46 OC. Problems of temperature stability arise from the rapid ^variation of elastic (K ) [7] and viscous (y) [8]

coefficients with temperature. ^For â ^variation ôf ^T ôf

1 OC, ^we êstimate â ^variation ôf ^D ⁼ K/y

^and

of 3 % ^that ^we have verified expe-

rimentally. Ôur measurements were done at 22 °C with a temperature accuracy better than 1 OC by circulating âir regulated ⁱⁿ temperature ⁱⁿ â ^closed volume surrounding the flow cell and the microscope through which observations âre made. In addition,

the poor chemical stability ^{of this} compound ^{causes a}

drift of properties ôver â period ôf ^several days;

consequently, ^data ^are ^taken ^{over a} restricted period

of time. The experiments ^are performed ^for increasing

values of N keeping ^a ^constant ellipticity ^E ⁼ Xo/Yo.

The value of the thresholds are determined by ^observ- ing ^the light diffracted by ^the grating ^formed by ^the

convective pattern using ^a laser incident at right angle ^on ^{the cell.}

We determine two thres- holds : N 1, the lowest-linear-one above which rolls ^are obtained continuously ^{and in} ^a reversible _way;

N2 ( > N1 ) ^where â ^two dimensional pattern, ^consist- ing usually ôf squares caused by secondary ^rolls ât right angles the initial _ones, appears reversibly.

shows deviation for small d or f ^{from the} hyperbolic

Fig. ^2.

dots (b) and squares (a)) ^and ^square pattern (open ^dots (b)

and squares (a)) ^threshold in coordinate (Xo, Yo) :

shape predicted by (2) : ^a cusp is observed for the roll threshold near the 45° line corresponding ^to ^the

value E ⁼ 1 where the shear becomes circular in the bulk of the cell.

The existence of two branches symmetrical ^with respect ^to this axis is obvious from symmetry ^conside- rations as is also substantiated by the observation of the direction of the rolls (Fig. 3) : in the lower octant

(E 1 ), the rolls form an angle g/ - 20° with the x axis nearly independent ^of E; ^the upper ^octant (E > 1)

is obtained by exchanging the role of x and y ^{in the} experiment, ^the angle ^is

Fig. ^3.

This experimental ^value observed also by [1, 3] ^{is much} larger than the theoretical one 4/ - ³⁰ (J. ^M. Dreyfus

and C. D. Mitescu [9] ^have introduced in the calcula- tion [2] convective terms which _may explain ^this disagreement). ^The existence of a quasi discontinuous

change ^of angle 4/ (Fig. 3) around the value E ⁼ 1

(circular polarization) ^is associated with the change ^of

symmetry ^{of the} hydrodynamic shear around this

particular ^value. ^It ^is ^thus reasonable to relate the cusp in the threshold ^curve ^to the effect of the dege-

neracy around the E ⁼ 1 value.

J. Sadik and coworkers [10] ^have recently proposed

an analysis ^{of this} cusp. We will not reproduce ^their argument ^which ^is ^based ^{on an} ^extensive development

x --+ y and 0 --+ 7r/2 - 4/) applied ^to the linear solu-

tion of the nematodynamic problem. We will write instead an « ad hoc » equation ^for ^threshold ^which reproduces their main result. The analysis leading

to the form (2) ^assumes that the ratios Xo/d, Yo/d,

which represent ^the normalized displacements ^{of the} plates ^are ^small. ^It is also the limit where the alternat-

ing ^director distortions are proportional ^to ^{the shear.}

For larger ^shear amplitudes, ^non ^linear contributions

already appear in the alternating distortion below threshold [2]. ^{In such} conditions, ^the ^threshold ^can ^be

written as a symmetric expansion ^of (2) in powers of

X o/d ^and Yold (formula ⁽⁵⁵⁾ ^of [10]) :

The absence of linear term comes from the invariance of the problem ^with changes ^{x - -} ^x, y - - y. The absolute value in the right hand side of (3) implies

that two solutions are met depending ^on ^whether Xo Z Yo. A ^{and B} ^are functions of the material

properties. În principle, they ^can ^be ôbtained ^{from the} complex ^linear analysis. Â determines the amplitude

of the cusp, B that of the _cusp angle ² ^0. ^*

From the study ^{of the} ^two branches of solution of (3) along ^{the line} Xo = Yo, ^one gets ^an angle (Fig. 2)

between the tangents given ^for large d and f, by (formula (57) ^of [10]) :

We have analysed ^the validity of the dimensional form (4) by studying the threshold curves for nematic films of different thicknesses (from ^d ⁼ ³⁰ ^to ¹¹⁰ pm)

at different frequencies (from f ⁼ ⁹⁰ ^to ⁵⁰⁰ Hz). ^The