Threshold conditions for hydrodynamic instabilities in two-frequency nematics

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Threshold conditions for hydrodynamic instabilities in two-frequency nematics

P.R. Gerber

To cite this version:

P.R. Gerber. Threshold conditions for hydrodynamic instabilities in two-frequency nematics. Journal

de Physique, 1985, 46 (11), pp.1865-1872. �10.1051/jphys:0198500460110186500�. �jpa-00210137�

Threshold conditions for hydrodynamic instabilities in two-frequency ^nematics

P. R. Gerber

F. Hoffmann-La Roche & Co. Ltd., Central Research Units, ^CH-4002 Basel, Switzerland

(Reçu ^{le 20 mars} 1985, revise le 7 juin, accepti ^{le 28} juin 1985)

Résumé.

Les équations électrohydrodynamiques pour ^une couche de nématique ^en configuration planaire

sont traitées dans le cas de petites déformations. On étudie les conditions pour ^une instabilité hydrodynamique : premièrement ^{dans le cas d’une} anisotropie diélectrique positive ^{sous la forme d’un} développement ^en petits ^vec-

teurs d’onde (déformation ^de longue période) ^et deuxièmement dans le cas de nématiques ^{à « deux} fréquences ^».

Pour une anisotropie positive ^{de la} conductibilité électrique, ^la déformation statique ^{de Frederiks est} supplantée

par ^une instabilité dynamique ^à période finie. Les instabilités ne se produisent ^{que dans une bande de} fréquence large ^d’à peine ^un facteur deux. Les conditions sont optimales pour ^un processus de relaxation de grande amplitude

et pour ^une anisotropie diélectrique négative ^{forte à} fréquence ^élevée.

Abstract

Electrohydrodynamic equations ^{for a} layer of nematic material with planar homogeneous alignment

are given ^for alternating applied voltage, ^{for the case} of small deformations. Conditions for hydrodynamic ^instabi-

lities are studied firstly ^{for the case of a} positive dielectric anisotropy ^{in a small} wavevector-expansion, ^and secondly numerically ^for two-frequency ^nematic materials in which the conductivity originates ^from orientational relaxation of polar molecules. It is found that, except ^for some cases of negative conductivity anisotropy the static Frederiks deformation can only ^{be overtaken} by ^a finite-(non-zero) wavevector dynamic instability. ^In two-frequency ^mate-

rials instabilities hardly ^{occur over a} frequency range of ^more than a factor of two. Best conditions are a high ampli-

tude of the relaxation process and large negative dielectric anisotropy ^at high frequencies.

Classification Physics ^Abstracts 61.30G201385.60P

1. Introduction.

Electrohydrodynamic instabilities in nematic layers

which are oriented homogeneously parallel ^{to the} plane of the substrates by surface forces have been studied for some time [1]. The observation by ^Wil-

liams [2] ^of regular patterns ^{which occur} in certain substances _upon application ^{of a} perpendicular ^electric

field have found a transparent explanation through

the work of Helfrich [3]. Heilmeier et al. [4] ^{found that}

at higher applied voltages ^turbulent patterns develop

which owing ^{to their} strong light scattering properties appeared ^{to be} promising ^for display-application.

Helfrich’s interpretation required ^an anisotropic ^elec-

trical conductivity ^which enabled the generation

of _space charges. ^{In most} experiments this conduction

was provided by ^ionic impurities in the nematic material. These impurities ^{in turn} hampered ^the application of the effect because they were a source

of electrochemical degradation. ^{Later on} de Jeu et al.

[5] discovered similar phenomena ^{in a so called two-} frequency material for which part ^{of the} parallel

ohmic conductivity ^was provided by orientational

relaxation _processes of long molecules. This showed,

in principle, ^a way of avoiding degradation.

In the present ^{work we aim at} exploring ^the

necessary conditions ^on the dielectric parameters ^of ion-free nematic material for the generation ^of hydrodynamic instabilities, by studying theoretically

the threshold conditions of the electrohydrodynamic equations. ^To this end the existing theories had to be extended While the treatment of Dubois-Violette

et al. [6] ^{allows for} alternating applied voltages,

it is restricted to infinitely ^thick layers. On the other hand the finite-layer treatments [7, 8] ^{are restricted}

to dc-applied voltage ^{in which case} the orientational- induced conductivity ^vanishes.

A further, ^more fundamental question regards ^the changeover from static deformation to hydrodynamic instability ^{which occurs in these} substances _upon

increasing ^the frequency ^{of the} applied voltage.

2. Basic equations.

There are three types ^of equations which govern the

electrohydrodynamic effects in nematics namely ^the

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198500460110186500

1866

torque equation describing ^the alignment ^{of the}

nematic director, ^the acceleration equation governing

the material flow and Maxwell’s equations ^which

determine the charge distribution. Derivations of these equations ^{as valid in a nematic} layer ^{have been} given in several places [1, 6, 7] ^{whence we restrict}

ourselves to writing ^them down for the case of a

layer of nematic substance which is homogeneously aligned ^{in the substrate} plane by rigid boundary

forces at both substrates. The director deformations

are assumed to be restricted to the plane given by

the layer normal and the substrate aligning ^direction.

Similarly ^{we assume that} the material and charge

flow directions are also restricted to this plane.

The z-axis of the coordinate system is normal to the substrates with the origin in the middle of the

layer. The substrate positions ^{are thus at z}

± d/2.

The x-direction corresponds ^{to the} alignment ^direc-

tions at the substrates.

The variable fields are 9, the angle ^{between x-axis}

and director, Vx and vZ, the components of the mass

velocity, p the pressure, q the charge density ^and

the electric field variables Px and P. which constitute the electric field as

With these definitions the torque ^balance equation

reads

where

where y, and Y2 ^are viscosity coefficients [1]. ^{With the} density ^{p and the} viscosity coefficients _al to a6 [1] ^the

acceleration equations ^read

which have to be supplemented by ^the continuity equation of the fluid which is assumed to be incompressible :

With the conductivities all ~ and _(J.1 Maxwell’s equations ^read

All these equations ^are derived under the assumption

that the variable fields are small which ^means that all terms quadratic ^{or of} higher order in the fields

can be neglected. ^This procedure ^{is exact} for the cal- culation of threshold conditions as long ^{as the insta-}

bilities under consideration develop continuously

from the undeformed homogeneous ^state.

These equations ^{have to be} supplemented by ^the

boundary conditions at the substrates, namely

3. Time dependence.

The externally applied field j2 ^{Eo cos mo t oscillates}

in time with frequency mo. In the present ^treatment

we will assume that a period io

2 nlcoo ^{is short} compared ^to characteristic times governing ^{the chan-}

ges in director orientation or material flow. However, the electric fields and charge ^{field are assumed to} respond with times comparable ^to To (conduction regime [6]). ^{Thus, we set}

Furthermore we restrict ourselves to stationary ^solu-

tions such that all fields _{9, p,} vj, qk, fli,k ^{can be consi-}

dered time independent ^{To obtain} equations ^for

these quantities, ^{we take} time-averages ^of equations (2.2) ^and (2.9) ^while equations (2.11) ^to (2.13)

each yield ^an in-phase equation, ^obtained by ^multi- plying ^{with cos} roo t ^and time-averaging, ^{and an} out-of-phase equation, ^obtained by multiplying ^with

sin wo t ^and averaging. ^{We are thus} left with ten

equations. Similarly ^the boundary ^condition requir- ing vanishing tangential electric field at the substrate

splits ^{into two} components ^for P.,, and lix,2.

There is little use in writing down all the equations

at this point ^{However, it} might be worthwhile men-

tioning ^{that a treatment} of the dielectric regime [6, 9]

would require ^to take q ^as time-independent (or slowly varying), while the director field _qJ and the

velocity and pressure fields would have ^to be taken

as time dependent, ^{in a form} analogous ^to equa- tion (3.1).

4. Further treatment : characteristic equation ^and boundary conditions.

The treatment of the ten equations proceeds by ^elimi- nating ^{variables and} taking spatial ^{Fourier trans-}

forms. Some details can be found in the Appendix.

Taking ^the viscosity coefficients _ai and _a3

(11 ⁺ y2)/2 ^{which seem to} be small in the few known cases, equal ^{to zero for the sake of} simplifying ^the

arithmetic somewhat, ^one eventually ^{arrives at the} following ^three equations :

in which we have set

From the k-dependent variables t/J, ^v and ^the original ^{variables are} calculated through

An important equation ^relates P2 ^{which occurs in the} boundary conditions with Pl and

For a nontrivial solution of equations (4. 1) ^to (4. 3) ^{one has to} require ^a vanishing determinant of the coeffi-

cients of _v, Pl and V/ which leads to the characteristic equation

1868

This fifth order polynomial in y ^{has one solution} yo

1 (due ^{to the choice} ai

0). ^In comparison ^{with the}

static case [7] (lim coo -+ 0) ^the degree ^{of the} polynomial ^is augmented by ^{one thus} providing ^an additional solu- tion to allow to fulfil the additional boundary condition for the out-of-phase electrical field.

The k,,

^{0 solutions obtained} by setting y

^{0 in} equation (4.14) give ^the threshold condition

This is precisely the result obtained by Dubois-Violette et al. [6] for the conduction regime.

Each root yi of equation (4.14) has with it a corresponding small-amplitude solution, ^the coefficients _vi flu and ^{of which are} calculated through equations (4.1) ^to (4. 3) up ^{to a common} factor.

To fulfil the boundary conditions, any linear combination of these solutions _may be chosen. Actually, ^there

are ten conditions which, because of the mirror symmetry ^{at the} x-axis of the problem, ^{reduce to} five each for

symmetric ^and antisymmetric solutions. These conditions again ^{have the} form of five homogeneous ^linear equa-

tions, having ⁱⁿ general ^{no solution} except ^for special values of the coefficients which give ^{zero value of their}

determinant. For the case of symmetric solutions which is most interesting ^{in the} present context, ^the determinant reads

where we have used the abbreviations

The values of the external field parameter ^{U for which} equation (4.16) ^is obeyed, give ^the required ^threshold

conditions.

5. Expansion ^{for small wave numbers} k,

For two-frequency nematics in which the dielectric

anisotropy changes sign ^to negative values with

increasing values of the frequency, ^a transition from

a static deformation to an electrohydrodynamic ^ins- tability ^{can occur for} increasing frequency. ^{Since the}

static deformation is a kx

^{0 solution, one} might

expect ^to gain ^some insight ^into the transition from static to dynamic instability by attempting ^{a small-} kx expansion of the above threshold conditions (4.16).

To perform ^the calculation, ^we restrict ourselves to vanishing magpetic ^field (h

0) ^and use as an

expansion parameter the dimensionless quantity

where A is the spatial period of the deformation.

Remembering ^{that the} parameter ^U diverges when kx

goes ^{to zero we set}

where the parameter

is just ^the square of the ratio between the applied voltage ^{V and a reference} voltage PB which is related to the Frederiks threshold voltage [1] ] VF by

Inserting (5. 2) into the characteristic equation ^reveals

that one solution, _{Y4, is of} order, - 1, while the remain-

ing ^{four are of} order (including yo = - 1). ^{A small}

calculation yields

Furthermore, ^{we have}

.

and for the solution y4

such that

Because of

the quantities C4 ^and S4 ^{are at most of order} çO.

Thus to order (° the determinant (4.16) ^reads

In this expression ^{the two} determinants contain only

the four solutions y°, ... Y3 of O(çO), and have in

general ^{nonzero values.} Because Y4 ^{is of} order equation (5.13) requires

This immediately yields ^from (4.17) ^and (5.12)

The threshold voltage for the lowest (n

0) ^{mode is}

thus given by

This expression shows that for the common nema-

tics which almost always obey _Q II > (J.1 the static deformation with ç

⁰ always ^{sets in before a} dyna-

mic deformation with small ç. ^{In other} words, ^{at the} changeover from static to dynamic instability ^{the i}

latter always ^{sets in with a} finite, ^non zero wave-

vector kx ^{as a} first order transition. In section 6 this will also be illustrated by ^{a numerical} example.

In contrast to ordinary ^nematics composed ^of elongated molecules, discotic molecules which show

a nematic-type order, ^{seem to} obey ^{the condition}

(J.1 > all II [10]. ^{Thus, for low} enough frequencies ^and

k 1

(J II I (J .1 ^{one would expect} that the usual

Frederiks transition is prevented by ^{the occurrence}

of a hydrodynamic deformation mode in such mate- rials. In addition, for suitable parameter ^{values one} may observe a second order transition from static to dynamic deformation.

6. Numerical calculations for two-frequency ^nema-

tics.

We restrict ourselves here to the case of a single ^relaxa-

tion _process at frequency Cùr in the dielectric constant

parallel ^to the nematic director and zero ionic conduc-

tivity

The crossover frequency uy is obtained from the con-

dition _y

0 (Eq. (6.1))

Our aim is mainly ^{to trace the} conditions for dynamic

instabilities when the dielectric parameters yo and

Ay vary. For this _reason, the elastic and viscous _para- meters are kept ^{fixed at the more or less} representa- tive values

With these material parameters ^the following ^results

have been obtained.

In figure ^{1 the} dependence ^{of the} instability voltage

is shown for varying ^wave number for several fre-

quency ^{values of the} applied voltage. ^{Above the cross-}

over frequency We the instability ^{sets in at a finite}

wavevector given by ^the voltage minimum. Below We ^a second minimum occurs at dl À.

^{0 which at}

first lies above the other but at a frequency Wx drops

below. At this frequency ^the changeover ^between dynamic and static deformations takes place. Upon

further lowering ^the frequency, ^{the now metastable} dynamic ^minimum disappears’ altogether. ^{The wave-}

vector of the dynamic instability ^decreases steadily

with decreasing frequency ^{to a finite non-zero value,}

where the dynamic ^minimum disappears. ^This picture

as well as the curvature at dl À.

^{0 for co} > we is in accordance with the result of the small wavevector

expansion of section 5.

The threshold voltage ^{as a} function of frequency

is shown in figure ^{2. The crossover} between static and dynamic deformation occurs at the frequency Wx’

somewhat below _Wc. Also shown is the wavevector of the deformation (as dl À). ^{With this value, one can,}

for comparison, ^{calculate a threshold as obtained}

1870

Fig. ^1.

Wavevector dependence of threshold voltage ^for

various frequencies ^{in a two} frequency material. For low

frequencies the minimal voltage ^{occurs at À.}

oo, which represents the static Frederiks deformation. For increasing frequency ^{a finite} wavelength hydrodynamic instability

takes over while for A

oo a local minimum is still present.

For w > Wc’ only ^the dynamic minimum remains. The minimal voltage instability ^{shows a} pronounced wavelength

variation with varying frequency (dashed curve).

by the kz

⁰ approximation of reference [6], ^which

is given ⁱⁿ equation (4.15). Clearly, this value is

expected ^to improve ^with increasing ^ratio dIA. ^For

the calculations of figure 2, ^this approximate ^threshold voltage ^{is low} by ^{almost a factor of two at} wx and by

some 30 % ^at wlw,,

^1.3.

Also shown on figure ^{2 are threshold} voltages ^for

a few nonzero values of the ionic conductance assum-

ing (Ji II I Ui.1

^1.5. As found in experiments [5],

the _range of frequencies for which the dynamic ^ins- tability ^{is observed} increases when the ionic relaxa- tion frequency

increases to values comparable ^to w,,.

For our purpose, the most relevant results are

shown in figures ^{3 and 4. For} these calculations the ionic conduction was again ^{set to zero and the dielec-}

tric parameters yo and Ay ^{were varied.} Figure ³

shows contours of constant threshold voltage V x

as observed at the frequency Wx where static and

dynamic instabilities exchange. The lowest values are

encountered for large Ay ^{and low} yo values. The fre-

Fig. ^2.

Frequency dependence of minimal threshold

voltages. ^The plott illustrates the fairly ^narrow frequency

interval in which hydrodynamic instabilities occur. Also shown (dotted curve) is the threshold voltage for the bulk

approximation (Eq. (4.15)) ^as calculated from the actual wavenumbers shown in the upper graph. ^{The dashed curves}

illustrate the enlargement ⁱⁿ frequency range obtained by ^a

finite ionic conductivity.

quency OJx hardly goes below 0.9 oj,,, (see ^Eq. (6.3))

such that it appeared ^not necessary ^to present ^a

,co.,/coc ⁱⁿ graphical ^form.

Of importance ^for possible applicability ^{of such}

materials is the _range in frequency within which

hydrodynamic instabilities can be generated ^without

the addition of ionic dopands. Figure ⁴ gives ^{a con-}

tour plot ^{for this} range. For this graph, ^the range ^was defined by ^the limiting frequencies OJx and a)hl the

frequency ^at which the threshold voltage had increas- ed to 3 V x. ^{It must} however be kept in mind that _coc may be a more practical ^lower range limit than _cvx, because at the more elevated voltages ^{used in} dyna-

mic scattering the static deformation _may again ^take

over for (o OJc. This is, however, ^{a minor correc-}

tion regarding the closeness of _(ox and _wc.

From figure ^{4 one sees that the} ranges stay usually

below a factor of two in frequency. ^The largest ranges

occur for large Ay ^{and low} yo values, i.e. in the same

region where the threshold is low.

Fig. ^3.

Contours of constant threshold voltage Vx ^{at the} changeover frequency between static and dynamic ^instabi- lity. The coordinates are the amplitude of the relaxation process Ay ^{and the} high frequency dielectric anisotropy yo.

Fig. ^4.

Contours of constant frequency interval for the

occurrence of dynamic instabilities in the (Ay, yo)-plane.

The interval is defined by Wx and the frequency Wh ^at which the threshold voltage has reached 3 Vx.

In conclusion, nematic materials with large ^orien-

tational relaxation effects in the parallel ^dielectric

constants may possibly ^{be used as} substances for

dynamic scattering ^without showing ^{the adverse} degradation ^effects usually observed in ionically conducting substances.

However, ^the range of frequencies ^{in which the} phenomenon ^is encountered is fairly ^{narrow. Fur-} thermore, ^{in order to} obtain low driving frequencies,

the relaxation frequency coo ^must be kept ^{low. This}

in turn leads to rather strong temperature dependences

of _coo [10] which would either limit the temperature

range of operation ^{or otherwise} require ^{a suitable} adjustment ^{of the} driving frequency ^{to the} display temperature. ^For materials with cvo/2 x = ^{1 kHz}

1

is, ⁱⁿ fact, very high.

The strong dependence ^{of the} wavelength ^{of the}

distortion on the value of the driving frequency

may possibly ^{be used to} produce gratings ^of variable, externally controlled period This would be in a range centred around a value given by ^the spacing ^{of the} liquid crystal ^cell.

Appendix.

Some of the tedious but straightforward arithmetical steps which lead from the basic equations ^{of sec-}

tion 2 to equation (4.1) ^to (4.3) ^{are sketched here.}

As mentioned in Section 4 we put ai

a3

0. The main steps include time averaging ^as described in Section 3 and spatial Fourier transformation by

means of equations (4.10) ^to (4.12). ^{The choice} (4 .11 ) ^{leads to a} single velocity transform vk ^{due to} equation (2.10). Similarly the number of transforms for the electric field (Eq. (4.12)) is reduced by (2.11).

Equation (2.2) yields directly (4.2). ^From (2. 8)

and (2.9) ^one eliminates _p and arrives at

Similarly (2.12) ^and (2.13) yield ^four equations (in-

phase ^and out-of-phase, ^{see Section} 3) namely

1872

Elimination of q1 ⁱⁿ (A. .1 ) ^{with the} help ^of (A . 3) ^leads

to (4.1). Elimination of 41 ^from (A. 3) ^and (A. 6)

and of 42 ^{from (A. 4) and (A. 5)} yields

From these equations (4. 3) is obtained by elimination of P2’ ^while (4.13) is identical with (A. 7).

References

[1] ^{For a review see} equation ^DE GENNES, ^P. G., ^The Phy-

sics of Liquid Crystals (Oxford, 1974) ^{or HEL-} FRICH, W., Mol. Cryst. Liq. Cryst. ²¹ (1973) ^187.

[2] WILLIAMS, R., ^{J. Chem.} Phys. ³⁹ (1963) ^384.

[3] HELFRICH, W., ^{J. Chem.} Phys. ⁵¹ (1969) ^4092.

[4] HEILMEIER, ^G. H., ZANONI, ^{L. A. and} BARTON, ^L. A.,

Proc. IEEE 56 (1968) ^1162.

[5] ^DE JEU, ^W. H., GERRITSMA, ^C. J., ^VAN ZANTEN, ^P.

and GOSSENS, ^{W. J.} A., Phys. ^{Lett. 39A} (1972) ^355.

[6] DUBOIS-VIOLETTE, E., ^DE GENNES, ^{P. G. and} PARODI, O.,

J. Physique ³² (1971) ^305.

[7] ^{PENZ, P. A. and} FORD, ^G. W., Phys. ^{Rev. A 6} (1972)

414.

[8] NANDY, P., ^Mol. Cryst. Liq. Cryst. ¹⁰⁴ (1984) ^281.

[9] HEILMEIER, ^{G. H. and} HELFRICH, W., Appl. Phys. ^Lett.

16 (1970) ^155.

[10] ^{DUBOIS, J.} C., HARENG, M., ^LE BERRE, ^{S. and} PERBET, J. N., Appl. Phys. ^{Lett. 38} (1981) ^11.

[11] GERBER, ^P. R., ^Z. Naturforsch. ^37A (1982) ^266.

[12] SCHADT, M., ^Mol. Cryst. Liq. Cryst. ⁸⁹ (1982) ^77.