Orientational optical non-linearity near the instability of nematic liquid crystal flow over an inclined plane

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Orientational optical non-linearity near the instability of nematic liquid crystal flow over an inclined plane

Yu. S. Chilingaryan, R.S. Hakopyan, N.V. Tabiryan, B. Ya. Zel’Dovich

To cite this version:

Yu. S. Chilingaryan, R.S. Hakopyan, N.V. Tabiryan, B. Ya. Zel’Dovich. Orientational optical non-

linearity near the instability of nematic liquid crystal flow over an inclined plane. Journal de Physique,

1984, 45 (3), pp.413-420. �10.1051/jphys:01984004503041300�. �jpa-00209771�

Orientational optical non-linearity ^{near the} instability

of nematic liquid crystal ^{flow over an inclined} plane

Yu. S. Chilingaryan, ^{R. S.} Hakopyan, ^{N. V.} Tabiryan ^{and B. Ya.} Zel’dovich (*)

Physics Department of Yerevan State University, 375049, Yerevan, U.S.S.R.

(Reçu ^{le 20} avril 1983, révisé le 7 octobre, accepté le 27 octobre 1983)

Résumé.

On étudie le problème de l’écoulement d’un nématique ^{sur un} plan incliné. Dans le cas où le directeur est ancré sur la surface du plan ^incliné perpendiculairement à la direction de la gravité ^et à celle de l’écoulement, ^on prédit ^une instabilité spécifique de l’orientation dont on calcule le seuil. On étudie la non-linéarité optique ^d’orien-

tation du nématique ^au voisinage du seuil de l’instabilité d’écoulement.

Abstract.

The problem ^{of the} gravity flow of nematic liquid crystal (NLC) mesophase ^{over an inclined} plane ^is investigated. ^{In the case when} the director of NLC is strongly ^{anchored on} the inclined plane ^surface perpendi-

cular to the gravitational force and flow direction, ^a specific orientational instability arises for which the flow threshold value is obtained. Orientational optical non-linearity ^{at the} threshold of NLC flow instability is investi-

gated.

Classification Physics ^Abstracts

47 . 20 - 61. 30G

1. Introduction.

The coupling, ^via viscous stresses, between the direc- tor n and the flow velocity ^{v, is one} of the factors which determines the original hydrodynamic properties ^of

nematic liquid crystals (NLC) [1-5]. ^{Due to this} coupl- ing, viscous flow such as simple shear flow or Poi- seuille flow can become unstable even for _very low

(10-3-10-4) Reynolds numbers. The instability phe-

nomena of these viscous flows were investigated experimentally ^and theoretically ^{in detail} [1-5]. ^In particular ^two types of instabilities are mentioned.

The first one corresponds ^{to a} distortion of the director which is uniform in the plane (no; vo) composed ^{of the}

undisturbed director and the flow direction. The second one corresponds ^{to a} so-called roll instability (with the roll axis parallel ^to the flow direction Oy).

Both instabilities arise at several critical values of shear rate and the threshold of the first type ^{of insta-}

bility is lower than the threshold of the second type.

The problem ^of ordinary viscous fluid flow over an

inclined plane is well known (see [6], § 5). If the boun-

dary conditions on the lower plane correspond ^to

strong anchoring ^{but the} upper surface of the fluid is free, ^the velocity profile has the form of the parabola

(*) ^{Institute on} Problems in Mechanics, prospect ^Ver- nadskovo, 101, 117526, Moscow, ^U.S.S.R.

and it is _easy to obtain the dependence ^{of the} steady

state thickness of the layer ^{on flow.}

The theoretically predicted ^and experimentally

verified orientational optical non-linearity ^of liquid crystals [7, 8] ^{is a matter} of intensive investigations ^at

present. ^In particular in several works (see [9, 10] ^and

the references therein) ^NLC optical non-linearity

near the Fredericks transition induced by ^a magnetic

or electric field is discussed, ^{as well as near} phase

transitions between liquid crystals modifications.

In the present paper ^we consider the problem ^of

NLC flow on an inclined plane assuming ^{the conditions}

of zero velocity ^and strong anchoring ^{on the} plane.

It is shown that for the definite geometry ^a specific

orientational instability ^{arises at some threshold}

value of flow. This threshold value and distortion

profiles ^are obtained for the case when the velocity

and director distortions are uniform in the plane (no; vo). ^The orientational optical non-linearity ^{at the}

threshold of the considered type instability ^{of NLC}

flow over the inclined plane is discussed.

2. Recording ^of boundary conditions.

A quantity Q (em 3 /s) ^{of the} liquid crystal, ^{which is}

contained in the _xy plane ^{and has a width} along ^the

x axis given by l, continuously ^flows along the y ^axis

which is inclined at an angle ^{a with} respect ^{to the}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01984004503041300

414

Fig. ^1.

Geometry : liquid crystal with thickness h flows

along the direction ey ^{over an inclined} plane which makes

angle ^a with the horizontal. The boundary conditions cor-

respond ^to no

ex.

horizontal. Then at not very high ^{flow velocities} (i.e. ^{for not} very high Q) steady liquid ^{flow with a}

thickness h occurs. If the boundary conditions on the inclined plane (plane ^{z =} 0) correspond ^{to a director}

orientation along _ey (no

ey) ^{or along ez} (no

ez),

then in the whole volume except ^{in a} boundary layer

of width ç, ^where

the molecules will be reorientated by ^the flow, making

an angle 0s ^{= arctg} (--’3/a2) with the axis ey ^{and stay} in the plane (y, z). ^{Here K is the} average elastic cons-

tant and q the viscous coefficient for the corresponding geometry.

We shall discuss in detail the situation when, ^on the boundary ^z

^{0 molecules are anchored} along

ex (n°

ex). ^{On the same} surface the condition on the flow velocity ^{v is v} (z

0)

^{0. The} boundary ^condi-

tions on the free surface z = h have the form

where Ki ^{are the Frank} constants; ^and (Ja is the sur-

face tension anisotropy. ^At the free surface the shear stress vanishes and this leads to the first condition in

(1). ^{The two} last conditions are obtained by ^consider- ring the addition to the surface part of the free _energy

depending ^on the orientation having ^{the form A =}

0.5 6a(nez)2 [12, 15, 16]. ca ⁼ 0 if the surface does ^not influence the director orientation. In the other limit (fa -> ^oo the surface strictly orients the director in the

layer plane (if (fa > 0) ^or perpendicular ^{to it} (if 7a 0).

Since we consider a situation where the undisturbed director direction is planar ^and homogeneous ^over

the whole volume of the NLC, ^{it is} necessary ^to consi-

der 6a > 0. In this case the orienting action of the free surface supports the director planar orientation.

Such a situation can be experimentally realized with the nematic PAA for example [17,18].

When the flow _{is very} weak the above mentioned orientation (n°

_ex) ^{will be} unperturbed ^{over the}

whole volume due ^to elastic effects. The viscous torque

acting ^on the director is zero and the viscosity, ^measur-

ed in this geometry, ^is fla ⁼ 0.5 a4. If we consider the inclined plane ^{to be} infinitely long, ^we obtain from

(A. 14) ^for velocity profile.

Then the _pressure distribution is

It is _easy to define the relation between the steady

state thickness h of LC and its quantity ^{of flow} Q, giving ^{on the} upper part of inclined plane :

All these expressions coincide with those for an ordi- nary liquid by ^a suitable definition of viscous cons-

tant na-

3. The system ^of linearized hydrodynamic equations.

Consider a light ^{wave with wave vector and} polariza-

tion vector lying ^{in the} plane (X, Z), ^{so that K =} (Kx, ^0, KJ, ^E

(Ex, ^0, EJ, and which is incident on

the described system. ^At sufficiently high ^{flow velo-}

cities the stationary profiles ^of velocity (2) ^{and direc-}

tor no ^are disturbed even without the _presence of an

electromagnetic ^wave.

Let us denote by ^{bn =} (0, _{n r} n _,), ^{bv =} (’Dx’ ’Dy, CU,- )

and P the orientation, velocity ^and pressure distur- bances. Equations coupling these disturbances can be obtained by linearizing (A. 12), (A. 14) ^{and the} equa- tion of uncompressibility OVi/OXi == Oi Vi ^{= 0. We}

carry ^out the linearization for the problem ^{which is} homogeneous ⁱⁿ the inclined plane, ^{i.e. we do not}

discuss the domain structure (010y

0), and the roll

instability (%x ⁼ 0). ^{Then we} obtain the following

system ^of equations :

where the symbol ^{* means} complex conjugation; ga = ell jj

-

E1 is the anisotropy ^{of the} mesophase ^dielectric

constant at the light frequency; ^{S is the} velocity gradient ^{of the} undisturbed flow along ^{z :}

fla’ fib ^are the Miesowicz viscosity coefficients :

The director perturbations ^{lead to the} field modification in the medium. However we can consider that in the

equations (4a)-( 4 f) E ^{const. because we are} interested in weak perturbations of the director proportional

to the light intensity. ^{In the same} approximation the director reorientation does not give ^{rise to an} Ey ^component

of the light field and therefore there is only y component ^{of the} torque acting ^on the director. Thus we only ^consi-

der (for ^the stationary problem) ^the z-dependence of the disturbances. In this _case, as for the simple ^{shear or}

Poiseuille flows problems, ^{one can show that} ’U’y ^{= ’U’z}

^p

⁰ everywhere, ^which corresponds ^{to a distur-}

bance of v(z) of the form bv ⁼ ex’U’x(z). ^{From the} equation (4c) ^and boundary conditions at z ⁼ 0 and z ⁼ h

we have :

With equation (6), equations (4a) ^and (4b) take the form :

Performing the transformation :

We obtain the dimensionless equations :

In this notation 4J., is determined from the equation :

where CÜx = h.1 (13 I crJx/K1.

Then, using ^{new variables U =} a2/3.(ii, ⁺ nz), ^{W =} a2/3.(ii, - ^{nz) and X}

^{all3 Z from (9) we obtain two}

uncoupled equations :

416

The boundary conditions (1) ^{in new} variables take the form :

where X1

4. Investigation ^{of flow} instability.

General solutions of equations (11 a) ^and (llb) ^{without an external} electromagnetic ^{wave are} expressed through

the Airy ^functions Ai(X ) ^and Bi(X) [20] :

where A1, A2, B1 ^and B2 ^are arbitrary constants. From the boundary conditions (12) ^we get ^four homogeneous equations for four unknowns A1, B1, A2 ^and B2. ^This system has non-trival solution when its determinant is ^zero.

Thus we shall find an equation ^{for critical} X1, i.e. for critical flow Qc :

where

D1, D2 ^{are constants} connected with Az(O) ^and Bi(O) :

Critical thickness he ^{and flow} Q, ^are expressed through ^{the root} Xl ^of equation (14) :

Consider the influence of the surface forces on the value of the threshold. We present ^the numerical solutions of equation (14) ^{for the cases} far ^{= 0} and Ea -+ ^{oo because we do not} know the value of NLC surface tension

anisotroPYO"a.If.Ea ⁼ 0 we get from ( 14) (F1 ⁺ D 1 ) (F2 ⁺ D 1 )

⁰ and, ^because F 1 (XJ is positive ^{for all} Xl > 0

we have F2 = - D1 and, therefore, X@ = ^{1.98. For} example ^from (16) ^{we obtain} h’ 3 ^{sin a = 23.5 gm and cor-}

respondingly the critical flow Q 0 "/l l-- ^{1.04 x} lO- 5 cm2 s-t for the parameters characteristic ^for LC p ⁼ 1 g/cm3, K1 ^{= 6 x 10-’} dyne, K2

^{4 x 10-’} dyne, Ila ^{= 0.25} poise, fib ⁼ 0.4 poise, a2 ^{= - 0.77} poise, CX3

0.012

poise [ 11 ].

When Ea -+ ^{oo from} equation (14) ^we get F1 ⁼ F2, ^therefore X c = ^{2.347. We have} h 3 sin a ^{27.85 um}

and correspondingly ^{for critical flow we obtain} Qoo/l ~ ^{1.7 x} 10- 5 cm2 s-1 for above mentioned parameters.

We obtained that the threshold in the case (Ja -+ ^oo is somewhat greater than in the case ca ⁼ 0. This is connected with the circumstance that the surface force supporting the director undisturbed orientation increases with the increase of the _a..

Knowing Xi ^{we can obtain} the disturbances profiles ^{near the} threshold, up ^{to an} indefinite constant multi-

plier. ^{To an} accuracy of two decimal places ^{we find :}

where A and B are arbitrary ^constants.

Let us note that at z ⁼ 0 X ⁼ X 1 ^{and at z = h X = 0.}

’Ù’ x ^can be found from equation (10) by integrating.

The profiles of disturbances nY’ iíz ^and ctJ x ^at Ea ^{= 0, ooare} given ⁱⁿ figure ^{2. As can be seen on} figure ^{2 the} profile of nz essentially depends ^{on the value} of ca ^{in contrast to the} quantities ny ^and ’BJx. ^{This is connected with}

the given boundary conditions (1). ^For obtaining ^the steady ^state amplitudes of disturbances above the threshold it is _necessary to solve a non-linear problem. ^{We will not} discuss such a problem ^{in this} paper.

5. Orientational optical non-linearity ^{near a flow} instability.

Dividing ^both parts ^of equations (l la) ^and (lib) by ^{I we} get equations whose solution is given by ^the Airy ^func-

tions Ai(X ), Bi(X ) ^and Gi(X ) :

’

where A1, A2, B1 ^and B2 ^are constants to be determined from the boundary conditions (12).

Near the threshold of the instability ^which would have the flow on the inclined plane ^{without a} light field, they have the form

where i

In particular ^{in the case} of Ea ^{= 0,} inserting numerical values for the parameters ⁱⁿ (18) ^and omitting higher ^order

terms, ^we have :

When X. 0 ⁰ the critical index for all four parameters AI, A2, B1 ^and B2 is ^{the same and is} equal ^{to - 1. For} example ^when Ma --> ^{oo we have :}

Fig. ^2.

Disturbance profiles nY’ nz ^{at I a}

^{0 (a) and ¿a -+ oo (b).} Disturbance profile ^of ’BJ x ^(c).

418

6. Discussion.

NLC flow on an inclined plane is considered in the present paper. The instability ^{of such a flow} corresponding ^to

director disturbance depended only ^{on the z} coordinate transverse to the layer ^is investigated. ^The physical

reason for the instability ^is the decrease in the effective viscosity ^{with a} decrease of the angle between the director and velocity [ 13], and this decrease is the same for positive ^and negative ^bn. Mathematically ^{this is} connected with

inequalities a2 0, a3 0, a2 I > I a3 I ^{which are valid for most of NLC.}

.

On the other hand, orientational forces in the absence of the flow try ^to hold the director strictly homoge-

neous over the whole volume. When the value of flow (or thickness) is above its critical value, the elastic forces cannot hold stable the homogeneous distribution of the director.

On the contrary ^{in the} problem ^with strong anchoring of the director on the inclined plane perpendicular ^to

the plane (no

ez) ^or along ^{the flow} (no

ey) the effective viscosity decrease takes place in the first order of bn,

i.e. it is sensitive to the sign of bn. As usual in these problems, ^the director disturbances 3n - V _appear at small

flows, i.e. the reoriention effect here has no threshold behaviour. Large ^director disturbances (3n - 1) appear here

approximately ^{at the same} value of flow at which the instability arises in the problem with no ⁼ ex’

Orientational optical non-linearity ^{near (but} below) ^the instability point of the described type is considered in the present paper. ^It is shown that perturbations of the NLC dielectric constant bgik

ga(n9 I n k + ni no ) critically ^{increases near} the threshold of the flow instability ^{as 3£ -} (Qc - Q )-1.

Let us give numerical estimations. For a ⁼ 0 i.e. in the absence of the flow, ^{for the wave incident} angle ^{in the}

medium 300 and for the LC-parameters h ^{£ 10-2} CM, Ba ⁼ 0.7, ^{K -} 10 - 6 dyne ^we obtain that the director reorientation to an angle - 0.1 rad in the _x, z plane ^is achieved for the incident beam _power density

P - 0.1 kW/cm2. ^{Such a} power density is available for usual ^c.w. lasers. The increase of a and the approach ^to

the hydrodynamic ^flow instability ^{threshold at a} fixed thickness leads to the decrease by ^{a factor} 6c/(6c " Q )

in the power density, required ^{for the} given director reorientation.

From our point of view it will be interesting ^to experimentally ^{find the} predicted instability, ^{to measure its}

threshold and to investigate orientational optical non-linearity ^{near the} instability point.

Acknowledgments.

The authors are grateful ^{to G. E.} Volovik and E. I. Kats for useful discussions.

Appendix.

DISSIPATIVE FUNCTION FOR UNIAXIAL NEMATICS TAKING ACCOUNT OF THE HYDRODYNAMIC MECHANISM OF RELAXATION.

In order to obtain the expression ^{for the} dissipative function of uniaxial nematics, ^{let us use the} properties which the material must have :

1. As free _energy F, dissipative ^{function R also must be even in n} (n ^{is NLC} director).

2. If the velocity gradients and the velocities themselves are not very high, it will be sufficient to let R depend ^on

terms of the order (ðVi/ðxk)2 ^{and less.}

3. R must vanish when the liquid uniformly ^{rotates as a} whole because it is clear that under such rotation there must not be _any friction in the liquid. Therefore the dissipative ^{function can} depend only ^{on the tensor} dij ^{and the}

vector X which become zero for such a rotation :

4. There must not be terms in R not depending on dij ^{and JM because at v = const. and dn/dt = 0, R must}

vanish.

5. The dissipative ^{function must be} non-negative and therefore cannot consist of terms linear in dij ^{and X.}

The most general form of the dissipative ^function (of uniaxial NLC) having the above mentioned properties ^is

For uncompressible ^NLC (did ⁼ 0) ^{we have}

Since n2 = 1 the Lagrangian ^{of NLC can} be written as :

The first term represents the kinetic energy density (p ^{is the} density); Fd-Frank’s ^elastic energy; F f-describes ^the

free _energy of interaction between the director and the external field; Fo-the internal free _energy depending ^on density. ^In (A. 4) ^we also introduce notations h-Lagraqge multiplier, P-hydrostatic pressure and u-the linear

displacement ^{of the} liquid particle, du/dt - ^v.

The equations ^{of motion can} be obtained from (A. 2), (A. 4) and from the variational equation :

Here qi ⁼ ni, Ui,

The obtained equation ^are called the Ericksen, Leslie and Parodi (ELP) equations. ^Before writing ^down

these equations ^{let us} give ^the expression for the viscous ^{stress tensor :}

Comparing (A. 6) with known (see ^for example [11]) expressions ^for U’ki ^we obtain the relations between the Leslie and Jli coefficients :

It is seen from (A. 7) ^{that the} Parodi relation (a6

a2 ⁺ a3 ⁺ a5 ) ^is only the result of dissipative ^function (A. 3 ) consisting ^of only ^five independent ^terms. Insertion of the sixth coefficient leads to the additional relation.

In order to define the inequalities arising because of the non-negativity ^of quadratic ^form (A. 3) ^{it must be}

reduced to the diagonal ^{form. For} this, ^{one must} separate ^tensor dij into irreducible components ^{in relation to the} one-parameter group of rotation around the axis ⁿ (compare ^with [14]) :

where do

Then for quadratical form (A. 3) ^{we have}

The following inequalities ^are obtained from this :

For obtaining ^{the fifth} inequality ^{let us write :}

Consequently ^{the fifth} inequality has the form :

JOURNAL DE PHYSIQUE.

T. 45, N° 3, MARS 1984

420

For writing ^{the ELP} equations ^{we will} suggest ^{that the} liquid ^{is in the} gravitational ^and electromagnetic ^fields

where g-the gravitational acceleration, _ga ^{= 9} ₁₁

_{G 1.} is the dielectric permittivity anisotropy. ^Then equations ^for

the director components ⁿ obtained from (A. 3) ^and (A. 4) have the form

Varying expressions (A. 3) ^and (A. 4) by ^{u and} using the formula (see [11]) :

We get the balance equation ^{of forces} acting ^{on a} liquid particle

Equations (A 12) ^and (A 14), along ^{with the} incompressibility condition div v

0, constitute a self-consistent system ^of equations ^for defining p, ^v and n.

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