Distortion waves and phase slippage in nematics

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Distortion waves and phase slippage in nematics

J.-M. Dreyfus, P. Pieranski

To cite this version:

J.-M. Dreyfus, P. Pieranski. Distortion waves and phase slippage in nematics. Journal de Physique,

1981, 42 (3), pp.459-467. �10.1051/jphys:01981004203045900�. �jpa-00209030�

Distortion waves and phase slippage ^{in nematics}

J.-M. Dreyfus

Laboratoire d’Hydrodynamique Physique, E.S.P.C.I., 10, ^rue Vauquelin, ⁷⁵⁰⁰⁵ Paris, ^France and P. Pieranski

Laboratoire de Physique ^des Solides, ^Bât. 510, Faculté des Sciences, 91405 Orsay, ^France (Reçu ^{le 9} septembre 1980, accepté ^{le 12 novembre} 1980)

Résumé.

On démontre que les ondes de distorsion peuvent ^se propager dans ^un milieu nématique ^infini lorsque

celui-ci est soumis au cisaillement polarisé elliptiquement. ^{Ces ondes sont} hélicoïdales et leur propagation ^cor- respond à la rotation de la distorsion hélicoïdale. On indique l’analogie ^{avec la} propagation des ondes de cisail- lement dans un fluide newtonien en rotation.

Dans une couche nématique d’épaisseur finie la rotation de la distorsion est observée en présence ^d’un champ électrique qui empêche la relaxation de la distorsion.

On développe l’analogie ^entre la distorsion dans une couche nématique ^{et le} paramètre ^d’ordre bidimensionnel d’un superfluide.

On observe l’enroulement et le glissement ^de phase.

Abstract.

It is shown that in the infinite nematic liquid, ^the distortion waves can propagate ^{when submitted to} the elliptically polarized ^shear deformation. These waves are helical and their propagation ^{can also be seen as a}

rotation of the helical distortion. The analogy ^{with the shear wave} propagation ^{in the} rotating newtonian fluid is indicated.

The rotation of the distortion is observed in a finite nematic slab submitted to an electric field in order to prevent the relaxation of the distortion.

The analogy ^{between the} distortion and the two-dimensional superfluid ^order parameter ^is developed. ^The phase winding ^and slippage ^{is observed} experimentally.

Classification

Physics ^Abstracts

61.30

67.40

1. Introduction. -The coupling, ^{via viscous stresses,}

between the director n and the flow velocity v, is one

of the factors which determine the original hydro- dynamic properties of the nematic liquid crys- tals [1, 2, 3]. Because of this coupling ^the viscometric flows such as simple ^{shear flow or} Poiseuille flow can

become unstable even for _very low Reynolds ^numbers.

The instability phenomena ^{of these} viscometric flows

were investigated ^{for several} years through experi-

mental measurements and theoretical calcula- tions [1, 2, 3].

One of the most striking ^{results was} that the insta-

bility threshold in viscometric flows is determined

mainly by the value of the Ericksen number Er [3].

It was pointed ^out by Pieranski and Guyon [4] ^that

this result is not général ; ^{the case of the} elliptically polarized flow shows some original ^{features :}

A. The instability threshold is not determined by

the Ericksen number but by ^another dimensionless number which takes into account the frequency

of the elliptical shear flow [4]. The detailed experi-

mental and theoretical investigations of these insta- bilities can be found in references [6] ^and [5].

B. In the present paper ^we pursue the conjectures

of reference [4] ^{and our aims are :} (1) ^to show theore-

tically ^{that the helical} distortion waves can propagate in an infinite nematic liquid ^{submitted to the} elliptical

shear flow and to point ^{out the} analogy ^{with shear}

wave propagation ⁱⁿ rotating ^fluid (section 2), (2) ^to investigate experimentally the effect of the elliptical

shear flow on the sinusoidal distortion in samples

of finite thickness (section 4), (3) ^to point ^{out the} analogy ^{with the} phase winding ^and phase slippage phenomena ⁱⁿ superfluids (section 5).

2. Helical distortion waves.

2. 1 UNPERTURBED

SHEAR FLOW.

Let us consider an infinité nematic

liquid initially oriented in z-direction (ninit ⁼ (0, 0, 1))

and submitted to the elliptically polarized ^{shear flow}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01981004203045900

460

described by ^the following displacement ^{field u :}

where Exo ^and E00FFo ^are respectively ^the amplitudes ^{of the}

.

aux ^au

déformations

ai’ az ; all other components ^{of the}

deformation tensor Ê

Vu are zero.

The velocity gradients ^Vu will induce the oscilla- tions of the director field via viscous torques. ^{In the} limit of small deformations (Eaxo, Eayo ¹⁾ the director components ^{are :}

where _a2 and _yi are the viscosity coefficient characte- ristic of the nematic compound ^used. Usually ^the

ratio (- a2 jyl) ^ranges between 0 and 1; ^the upper limit corresponds ^{to the most common cases.}

2.2 WAVE PERTURBATION.

The altemating ^flows (1) and the distortions (2) ^{are the} background ^{on which}

we superpose the distortion waves

of small amplitudes (nxo, nsyo Exo, eayo ^{« 1).}

The frequency mi 1 is now supposed ^{to be much}

smaller than w, so that the perturbation ns may be considered as quasi-static. Because of the coupling

between the flow and director, the distortion (3) ^will

be accompanied by ^{the flows of the same} frequency w1 : ^:

The total distortion n ⁼ na + nS and flow v = va + vS

must satisfy the Leslie-Ericksen equations. ^{Due to the}

fact that _cal was supposed ^to be much smaller than cv

these equations may be averaged ^{over the} period

2 nlw. ^This averaging procedure ^is presented ^else-

where [5]. ^For the purpose of the present calculations

we report only ^the final form of the averaged equations (eqs. (3.18), (3.19), (3.21), (3.22)) ^{in reference} [5].

where

a’s are the usual definitions of the viscosity ^coeffi-

cients [3] ^{while K} is the Frank constant in the approxi-

mation K11 1 ⁼ K22

K33 ^{= K} (1).

From equations (5a) ^and (5b) ^we get :

Substituting equations (3), (4) ^and (6) ⁱⁿ equations (5c) ^and (5d) ^we get ^the following system of the homo.

geneous linear equations ^{for the} amplitudes nsXO ^and nsyo :

2.3 DISPERSION RELATION.

The condition that the determinant of the equations (7) ^{must be zero} gives ^the dispersion equation :

The imaginary part imi ^{of the} frequency wi represents the damping factor due to the elasticity ^{K. The real}

part wi(= ± C03A9) determines the frequency ^{of the}

wave and is of the order of magnitude ^{of Q} (defined above) multiplied by the factor C which depends ^on

the viscosity coefficients of the nematic material.

Using ^{the same} values of the viscosity coefficients as

that used in reference [5] ^{we calculate}

and justify ^{the initial} approximation ^{of the} quasi-

static perturbations (mi ^« w).

(1) ^{As observed} by ^{one of} referees, ^{in fact} only ^{one elastic constant}

(K33) is involved in the case of the helicoidal distortion waves of

small amplitude.

2.4 HELICOmAL SOLUTIONS.

Using ^the dispersion

relation (8) ^we obtain from equations (7) ^the following

relation between the distortion amplitudes :

If, arbitrarily, ^{we set} nxo ^{to be a} real number _no, then the final expression for the distortion waves will be :

The solutions corresponding ^{to + and -} signs ^{in the}

above equations ^are nothing ^else but the left and right

hand winded helices rotating ^{around z} axis with the

angular frequency ^{C03A9 in the same} direction. This helical type of distortion is visualized in figure ^1.

Fig. ^1.

The helical distortion waves. Their propagation ^{can be}

seen as the rotation of the distortion as a whole.

From the equations (4) ^and (6) ^we deduce that the shear waves associated with the helical distortions are

helicoidal as well. However there is a phase lag ^between

the distortion and the flow (given by equation (6))

such that the shear avs/oz ^{is oriented at} right angle ^to

the distortion ns at each point of the helix. Its effect will

only ^{be a rotation} of the distortion helix with the

angular velocity ^{CQ. In} other words ^{we can} _say that because of this phase lag the distortion and the asso-

ciated flow are self-consistent.

2.5 ANALOGY WITH SHEAR WAVES IN ROTATING FLUID.

In the newtonian fluid at rest the shear

waves cannot propagate because of lack of the restor-

ing ^stresses other than the viscous ones which are

purely dissipative. However, ^{if we} suppose that the fluid rotates as a whole with respect ^to the inertial frame of reference, then the Coriolis forces must be

taken into account. Their role is to provide ^{the reac-}

tive restoring ^stresses necessary for propagation.

Following ^reference [7] ^{let us} suppose that the fluid is rotating ^{as a whole around the z axis with the} angu- lar velocity ^{co. Then let us consider a} perturbation having the form of a transverse shear wave :

where the velocities _Vx and vy ^are taken in the frame of reference (x, y, z) rotating with the fluid.

The perturbations (11) ^must satisfy the Navier- Stokes equations

which in the rotating coordinates

are :

where v is the dynamic viscosity of the fluid. The right

hand sides terms are the Coriolis accelerations.

Equations (12) ^are analogous ^{to the} equations (7) :

-

the velocities v correspond ^to the distortions ns,

-

the Coriolis forces in equations (12) ^are perpen- dicular to the velocities v, while in equations (7) ^these

were the viscous torques ^{which were} perpendicular

to the distortions ns.

The dispersion ^relation obtained from equations (12)

is analogous ^{to the} previous ^one (eq. (8)) ^{but the}

orders of magnitude of the factors which are involved

are quite different :

-

The damping factor in the case of the distortion

waves was determined by ^{the ratio} K/y, ^{which for}

MBBA is of the order of 5 x 10-6 cm2/s ; ^{in the}

newtonian fluid such as water the dynamic viscosity

v

ttlp is of the order of 10-2 cm2/s.

-

The angular velocity CQ, ^{in the case of the} distortion _waves, is only ^a small fraction of w, ^while

the inertial shear waves rotate with the angular velocity ^{2 w.}

3. Experimental.

^3. 1 PRODUCTION OF THE SHEAR FLOW.

The experimental ^set up which we used to

produce ^the elliptical ^{shear flow was} described in details in reference [6]. Figure ² depicts ^it only ^sche- matically. The nematic sample (MBBA) ^{is oriented} along the vertical axis z between two horizontal glass plates. ^Each glass plate ^vibrates independently ^{with the} frequency 03C9 in the horizontal plane ; ^one plate ^{in the}

x-direction and the other ^one in the y-direction.

A phase difference of n/2 is introduced between two vibrations. The respective motions of the plates ^then

can be written as :

462

Fig. ^2.

Creation of an elliptical shear in the nematic slab by

the combination of two perpendicular linear shears. These linear shears are obtained by ^the displacements ^{of the two} glass plates

in perpendicular directions ; - 03C9/ 203C0~ ^{203C0 - 200-500 Hz, xo and yo, the}

amplitudes ^{of the} plates displacements ^are of order of a few microns.

The actual motion of the liquid ^{in the} gap of thickness D between the limit plates depends ^on the viscous

penetration depth :

where _{p - 1} g/cm3 is the nematic density ^and

For 03B4 « D the shear waves are damped ^{near each} plate ^{and do not} interfere. For 03B4 » ^D the damping

can be neglected and the vibrations of the plates ^create

two independent linear shear deformations of ampli-

tudes :

which, ^when superposed, correspond ^{to the} ellipti- cally polarized ^shear deformation described by ^the equations (1) and visualized in figure ^2.

The inequality ô » ^{D limit} the upper range of

frequencies ^{to 03C9 « 2} ~bl(pD2) ~ ¹⁰⁴ Hz for the sam-

ple of thickness D ⁼ 100 pm. ^{Due to} the inertia of the moving parts ^{in the} experimental ^set up, the

frequency of excitation was additionally ^{limited to}

500 Hz so that in all experiments reported ^{in this}

article the deformation in the sample ^{can be well} approximated by equations (1).

3.2 CAN THE HELICOIDAL DISTORTION WAVES BE

OBSERVED ?

The necessary condition, which must be satisfied in order to observe effectively the rotation of the helical distortion is that the damping ^factor

Kq2/yl (eq. (8)) should be smaller than the frequency

of rotation CQ :

The minimum value of the damping ^factor (in ^{the case}

where no external fields are applied) ^is imposed by ^the

thickness of the sample D ; ^{the wave} vector q ^{= 2} ni À.

cannot be smaller than 2 niD. ^The frequency ^CQ

on the other hand is limited for two reasons : (1)

The amplitudes of the shear deformations must be small enough ^{in order to} satisfy the initial approxi-

mations of the theory (Exo, Eayo ^{« 1) and (2) the exci-}

tation (- ^{Q as defined in reference} [6]) ^{must be small} enough ^{in order to} avoid the flow instabilities. The

experimental results in reference [6] suggest ^{that the} instabilities cannot develop ^{for :}

The superposition of ( 16) ^and (17) ^{leads to the} following inequality

The fact that the _upper and lower limits in this in-

equality overlap slightly suggests ^that the direct obser- vation of the distortion waves is rather beyond ^the possibilities ^{of the} unmodified experimental ^set up.

3.3 EFFECT OF THE ELECTRIC FIELD.

The _way to avoid the difficulty ^{is to} prevent the relaxation of the distortion by ^the application of electric field parallel

to the z axis. In MBBA a sinusoidal distortion (Fig. 3) :

develops when the electric field larger than the Free- dericksz threshold Ec ^is applied. Following ^{this idea}

the experimental ^set up ^was modified. Using ^the

limit glass plates precoated ^{with the} transparent ^InO

Fig. ^3.

Definition of a two-dimensional order parameter ^above

the Freedericksz transition.

electrodes, the electric field was applied and induced the permanent distortion in the sample (2).

3.4 INFLUENCE OF THE LIMIT CONDITIONS.

The

homeotropic anchoring conditions (O(z = +D/2)=0)

at the limit plates impose ^the sinusoidal shape ^{of the}

distortion (19), the calculation of section 2 could

apply nevertheless to such a distortion because it can be

represented ^{as a linear} combination of two helical distortions of the opposite chirality. Consequently,

the sinusoidal distortion (19) ^{could be} expected ^to

rotate under the elliptical shear in the same way ^as the helical distortions (Fig. 1).

Unfortunately, ^the nonslipping boundary ^conditions

for the velocities v’(x(z

^± D/2) ⁼ 0) ^{are not compa-}

tible with the shear wave pattern (eqs. (4) ^and (6)).

Therefore there exists a boundary layer ^{close to each} glass plate ^{and the} theory developed ⁱⁿ chapter ^{2 is} only ^an approximation ^{to the real} experimental

situation. In particular, ^the angular velocity ^{of the} rotating distortion ^can slightly differ from the pre- dicted one (which equals ^to CQ) by the numerical factor C ’(03C9’1 ⁼ C’Q).

3.5 OBSERVATION OF THE DISTORTION IN THE POLA- RIZING MICROSCOPE.

The distortion in nematic slab is _easy to monitor between crossed polarizers

with the microscope. ^The interferential image ^allows

the straightforward determination of both the phase ^(p

and the amplitude ^cos 03B8m (Fig. 3) :

-

isoclines indicate the isophase ^lines (modulo n/2) whére the director n 1- is parallel ^{to the} polarizer

or to the analyser,

-

isochromes indicate the actual amplitude ^{of the}

distortion.

Let us consider ^{as an} example ^the singularities

characteristic of the Freedericksz transition in the

homeotropically ^oriented sample ^{known as umbi-}

lics [8] (Fig. 4). These defects ^were observed to occur

in our samples spontaneously. ^{In the} polarizing microscope, they appear ^as black crosses (Fig. 5) ^as expected ^{from the} configuration ^{of the} corresponding

distortions (Fig. 4) (see also section 4).

Fig. ^4.

^The three-dimensional structure of umbilics s

+ 1 and s

1. The black lines are parallel ^to the director. The

arrows represent ^the projection of the director in the mid-plane.

Fig. ^5.

Photographs ^{of a} pair ^{of umbilics s}

^{+ 1 and s}

1 observed in the polarizing microscope. ^{Under the} elliptical shear, ^the umbilics rotate in opposite directions. The photographs ^are separated by ^{time intervals} of a few seconds (see text).

4. Evidence for the rotation of the distortion.

The first evidence for the rotation of the distortion

(predicted in section 3.4) ^is provided by ^{the behaviour}

of the pair of umbilics (+ ^{1 and -} 1) ^{when the} elliptical ^{shear is} applied ^{to the} sample. ^The figures 5a, b and c show the configuration of isoclines at three différent times ta ^{= 0 s,} tb ^{= 5} sand tc ^{B= 10 s.}

It is evident from these photographs that the black

crosses characteristic of the umbilics are rotating :

The _{upper cross,} which corresponds ^{to the s = + 1}

umbilic rotates counterclockwise whereas the lower umbilic (s

1) ^{rotates clockwise.}

Let us consider more in detail the transformation of the umbilic s ⁼ + 1. The figures 6a-6h show how

e) ^The electrohydrodynamic instabilities are avoided because of high frequency (- ¹⁰⁴ Hz) and low value (- ⁵ V) ^{of the} voltage

used in experiment.

it transforms under the uniform clockwise rotation of the distortion. Each of the eight configurations ^i$

unique, nevertheless the corresponding ^{isoclines have}

all the same shape ^{of a cross.} During ^{the successive}

Fig. ^6.

Eight configurations ^taken by the umbilic s

+ 1 under the elliptical shear. The arrows are the directions of nl in the

mid-plane ^of the cell. The elastic energy varies during ^this cycle.

464

transformations the cross is rotating ^{in the counter-}

clockwise direction.

The detailed observation reveals that the angular velocity ^{of the cross} corresponding ^{to the s = + 1}

umbilic is not uniform. This is probably ^{due to the} anisotropy of the elastic constants :

and can be explained ^{as follows :}

The set of the eight configurations { ^{a, ...,} h } ^can

be divided in three subsets {a, ^e }, { b, d, f, h } an4 {c, g} ^{with different elastic} energies ^{E. The} inequality (20) suggests ^{that :}

The energy differences result in elastic torques ^which slow down or accelerate the rotation of the cross.

The behaviour of the umbilic s ^{= -} -1 is different.

The hyperbolic configuration (Fig. 4b) ^{is not affected} by ^the (clockwise) rotation of the director other than

as a uniform solid body rotation of the whole configu-

ration (in the clockwise direction). ^{In the} microscope

this is seen as a uniform rotation of the isoclines.

The angular velocity of the umbilics was observed to be dependent ^{on the} amplitude ^and sign ^{of the} quantity ^xo yo m ^as expected ^{from the} relationship :

5. Phase winding ^and relaxation.

The successful

experimental evidence for the rotation of the distortion under the action of the elliptically polarized ^{shear flow} provided ^{us with a basis on which we} developed ^{a series}

of experiments which have an intrinsic value as a way to visualize the winding and relaxation phenomenon

of the two-dimensional order parameter.

5. .1 INTRODUCTION OF THE TWO-DIMENSIONAL ORDER PARAMETER. ANALOGY WITH SUPERFLUIDS.

The dis- tortion (19) ^is characterized completely by the pro-

jection n 1. of the director in the plane (x, y) (Fig. 3).

From the formal point of view the vector n 1. ^can be considered as a two-dimensional order parameter

(j n 1. ^{1 = cos} Bm, cp) analogous ^{to that of} superfluids

or supraconductors. ^In superfluids, ^for example, ^the

order parameter being ^{a wave function}

cos 0. ^{in the} present ^{case will} correspond ^{to the} ampli- tude 1 03C8| and the azimuthal angle cp ^to the phase ~.

Following ^this analogy the umbilics (Fig. 4) ^are equi-

valent to vortices and antivortices in superfluids. ^It

must be emphasized ^{that the} important experimental advantage of the nematic liquid crystals ^{is the} straight-

forward visualization of the order parameter n~ ~

by ^the microscopic observation between crossed

polarizers (Sec. 3.5).

5.2 GENERAL PROPERTIES OF THE NONSINGULAR PHASE FIELD.

The general expression ^{for the} phase

field in the nematic sample ^is given by

where C’ is the numerical factor defined in section 3. 4.

Experimentally, ^the interesting characteristics of the phase ^{field, which can be observed, are the} velocity

of isoclines and their density. The isoclines being

defined as the lines of the constant phase, ^the phase

variation Vqp.ôr, produced by ^the displacement ôr,

must be annihilated by ^the phase rotation C’ 03A9 bt

during ^{the time interval ôt :}

C’ Q(x, y) ^{bt +} V(p - ôr

^{0 .} (24) C’Q(x, y) ^{ôt +} V~.ôr

^0. (24)

From this relation, ^the velocity v.1, measured in the direction perpendicular ^to the isoclines lines, ^{can be}

determined as follows :

where _a~, the unit vector perpendicular ^{to the} isoclines,

can be calculated as

The density of isoclines is directly proportional ^to

the modulus of the phase gradient :

In the case when the gradient ^{of the} angular velocity q

exists across the sample, the isoclines density ^will

vary in time. Such a situation, possible ^{to realize} experimentally, ^{as we will see in the} following, ^is expected ^to give ^{rise to the} phenomena ^{of the} phase winding ^and relaxation.

5. 3 EVIDENCE FOR THE PHASE WINDING.

The

angular velocity 03C91’ = ~ ^{= C’ 03A9 is a} function both of the material constants such as viscosities a’s and y 1

and of the quantity xo yo wlD2 which is determined by

the characteristics of the shear flow apparatus. ^In the _case, when the thickness D varies across the sample,

the angular velocity ^{C’ Q is a} function of the position

in the sample :

A simple configuration ^to realize is that of the wedge,

where the limit glass plates ^{are not} parallel ^{but form}

an angle ^a. Then, ^{we can write :}

where D * is the thickness of the sample ^{for x = 0} (Fig. 7), ^{D is} supposed ^{to be a} function of x only.

Let us suppose ^now that at time t ⁼ 0 the order parameter nl ^~ is uniform in the sample ^and parallel ^to

the x axis (cp(x, ^y)

0). According ^{to the} equation (27)

at time t the density ^{of isoclines} (measured along ^the

x axis) ^{will be :}

while their velocity ^as calculated from equation (25) ^{is :}

Fig. ^7.

Wedge parallel ^{to the} y axis between, ^the glass plates.

We observed directly, using ^the polarizing microscope (and simultaneous recording ^{on the} photographic film), the transformation of the isoclines configuration

under the action of the elliptical shear. The following

sequence of ^{events was} observed :

-

At time t ⁼ 0, ^the homogeneous aspect ^{of the} sample ^was indicative of the uniform orientation of the order parameter nl.

-

After the application ^{of the} elliptical ^{shear the}

isoclines started to appear ^at the thin edge ^{of the} sample

and to move in the direction of the increasing ^thickness (x axis).

-

The formation ^{of the new isoclines was more} rapid than their evacuation and as a result the isoclines

were progressively accumulated in the sample (Fig. 8).

Fig. ^8.

Photograph ^{of the} wedge shaped ^{cell as observed between}

crossed polarizers. ^{The black lines are} isoclines. Several umbilics disturb the isoclines alignment. The isoclines density ^decreases

in the x direction, ^{as the thickness increases.}

-

The density ^{of isoclines was observed to increase}

at a given point accordingly ^{to the formula} (30).

Also, ^{at a} given ^{time t this} density ^{was observed to}

decrease with the coordinate x (as expected ^from

eq. (30)).

-

Their velocity ^of propagation ^{at a} given point

slowed down with time ^as the isoclines ^{were accu-}

mulated (as ^indicated by ^eq. (31)).

5.4 PHASE RELAXATION.

The phase winding,

observed as an accumulation of the isoclines, ^was expected ^{to be} counteracted by ^{the elastic} torques, which develop ^{in the} sample ^{due to the} large phase gradients. ^One possibility ^{was that for some critical} density ^of isoclines these elastic torques ^{could be} strong enough ^{in order to} prevent ^{the further} phase winding.

Experimentally, the relaxation phenomena through

the emission and annihilation of the umbilics pairs

occurred before such a large ^isoclines density ^was

reached. Let us consider as an example ^the photograph

of the isoclines configuration represented ⁱⁿ figure ⁸

and let us focus our attention on the set of three umbi- lics A, B and C. The umbilics A and B are connected

by four isoclines. For the pair ^{B and C one could} say that four isoclines are missing. During ^the experiment

the connected pair ^{was observed to behave as} though

these were some effective attraction between them.

On the contrary, ^{in the case of the} pair ^{B and C some}

effective repulsion ^{was observed.}

The separation ^of the umbilics B and C leeds to the relaxation of the 2 n winding in the order parameter field as it is visualized schematically ⁱⁿ figure ^{9. The}

attraction between the umbilics A and B results finally

in their annihilation, ^and again ^{in the} relaxation of a

2 Te winding of nl.

Fig. ^9.

^Phase slippage mechanism. The attraction between the connected pair of umbilics results in their annihilation and relaxation of 2 n winding ^{in the} phase ^field.

The observation of the phase relaxation reinforces the analogy ^with superfluids. ^The phase ^relaxation by ^{motion of} singularities is known in fact as phase slippage mechanism and was proposed by ^{P. W. Ander-}

son [9] ^{in the case of} superfluids. ^{As far as the} topolo- gical features of the order parameter ^{field are con-} cerned, ^the phase slippage ^mechanism applies ^{to the}

present ^case.

The phase slippage ^{must be} preceded by ^{the emis-}

sion of the singularities. ^{In the case of the} superfluids

the vortex pair ^{emission mechanism was first} proposed theoretically by Langer and Fisher [10].

The nematic liquid crystals ^{offered us the} possi-

466

bility ^{to observe} experimentally the emission of umbilics step by step ^as it is shown in the _sequence of nine photographs ⁱⁿ figure ^10. Following ^reference [10]

the critical phase is that in which the order parameter

amplitude ^vanishes identically ^{at one} point ; ⁱⁿ photo- graph 10c this is seen as the coincidence of two iso-

clines which is possible only for 1 n~ 1 1 - ^{0. The next six} photographs ^{show the} separation of the umbilic pair.

We do not show the detailed schematic of the phase

field transformation (like ^{that in} figure 6) ^{because it}

would be exactly ^{the same as the} figure ^{1 in refe-}

rence [10]..

Fig. ^10.

Emission and separation of the umbilics pair.

6. Conclusion.

As compared ^{with the} previous developments ⁱⁿ the field of the hydrodynamics ^{of the}

nematic liquids ^the elliptically polarized shear flow appears ^{as a} unique ^case.

The phenomena ^{of :}

(1) ^{the new} convective instabilities [6, 5], (2) the helical distortion _waves,

(3) ^the phase winding ^and slippage

are original. ^{In all three cases the} analogies ^{were made}

with the phenomena occurring ^{in other} systems ^such

as :

(1) two-dimensional melting [6], (2) rotating ^{newtonian fluids,} (3) superfluids.

We believe that they ^{can be used as a} basis for the next both experimental ^and theoretical developments.

A number of new experiments ^{can be conceived} by ^the

careful modifications of the boundary conditions : The geométry of the electrodes is one of the _para*

meters which can be modified easily.

The geometry ^{of the} gap between the limit plates

can also be varied to some extent.

Finally, ^the possibility of the modification of the material constants (by ^{the use of différent} compounds

and temperature regulation) ^{should be examined.}

Acknowledgments.

^We greatly acknowledge

Professors E. Guyon, ^A. Libchaber, ^{and B.} Castaing

for valuable discussions conceming ^the analogy

with superfluids. ^{We take the} opportunity ^{to thank}

Mr. M. Clement for his constant assistance during

the realization of the experimental ^set up. One of us

(P.P.) ^is grateful ^{to Professor R.} Meyer ^{for his invita-}

tion to Brandeis University, where thanks to Pro- fessor C. Young ^{and Mrs M.} Meyer ^the manuscript

was critically reviewed and typed.

References

[1] LESLIE, F. M., Theory ^{of flow} phenomena ⁱⁿ liquid crystals,

Adv. Liq. Cryst. (1978).

[2] JENKINS, ^J. T., ^Ann. Rev. Fluid Mech. 10 (1978) ^197-219.

[3] DUBOIS-VIOLETTE, E., DURAND, G., GUYON, E., MANNEVILLE, P. and PIERANSKI, P., Liquid Crystals, ^{Solid State} Phys.

Suppl. ¹⁴ (1978) ^147.

[4] PIERANSKI, P., GUYON, E., Phys. ^{Rev. Lett. 39} (1977) ^1280.

[5] DUBOIS-VIOLETTE, E., ROTHEN, F., J. Physique ³⁹ (1978) ^1039.

[6] DREYFUS, J. M., GUYON, E., submitted to J. Physique.

[7] BATCHELOR, ^G. K., An introduction to fluid dynamics (Cam- bridge ^Univ. Press) 1967, p. 551.

[8] RAPINI, A., J. Physique ³⁴ (1973) ^629.

[9] ANDERSON, ^P. W., Rev. Mod. Phys. ³⁸ (1966) ^298.

[10] LANGER, Y. S. and FISHER, ^M. E., Phys. ^{Rev. Lett. 19} (1967)

560.