DYNAMICS OF ELECTROHYDRODYNAMIC INSTABILITIES IN NEMATIC LIQUID CRYSTALS

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DYNAMICS OF ELECTROHYDRODYNAMIC

INSTABILITIES IN NEMATIC LIQUID CRYSTALS

I. Smith, Y. Galerne, S. Lagerwall, E. Dubois-Violette, Geoffroy Durand

To cite this version:

JOURNAL DE PHYSIQUE Colloque CJ, supplkment au no 3, T o m e 36, Mars 1975, page Cl-237

DYNAMICS

OF ELECTROHYDRODYNAMIC INSTABILITIES

IN NEMATIC LIQUID CRYSTALS

I. W. SMITH, Y. GALERNE, S. T. LAGERWALL, E. DUBOIS-VIOLETTE and G. D U R A N D Laboratoire de Physique des Solides (*), UniversitC Paris-Sud, 91405 Orsay, France

R6sum6.

-

Nous Btudions le problkme des instabilites 6lectrohydrodynamiques dans les cristaux liquides nematiques en orientation planaire, dans le contexte du modkle une dimension propose par Dubois-Violette, de Gennes et Parodi. Dans la premikre partie de cet article, nous r6solvons analytiquement les equations du mouvement pour un champ Clectrique applique E dont la depen- dance temporelle est une onde carree de frkquence v. Notre solution couvre tout le domaine des paramktres du materiau, y compris le cas ou I'anisotropie dielectrique est positive ou negative, et est valide pour 0 5 v 5 10 kHz pour un materiau typique. On obtient ainsi une determination de la courbe du seuil d'instabilite qui montre en particulier que le seuil du regime de conduction est une courbe fermee rtgulikre dans le plan E - v, et dont la forme depend du temps de relaxation diklec-

trique z, du temps de relaxation viscoelastique T H et du paramktre de gain de Helfrich l 2 . Nous trai- tons le cas pour toute epaisseur d'echantillon d et nous montrons que pour des 6chantillons suffisam- ment minces tels que T H ( C C d2) < 2[2/(12 - I), la boucle du seuil du regime de conduction rktrecit

et disparaft, le regime se prolongeant ainsi jusqu'a v = 0. Nous traitons aussi, en une-analyse trks simple, le cas des non-linearit& et de la saturation qui apparaissent quand le champ applique est au-dessus du seuil. Nous somrnes alors en mesure d'expliquer le mode du riseau ci pas variable (qui

a kt6 observe expdrimentalement), pour lequel la largeur des domaines est inversement proportion- nelle au champ applique et peut varier d'un facteur dix au moins. Nous calculons aussi la vitesse de croissance (ou de decroissance) s d'une oscillation au voisinage du seuil, I'amplitude variant lente- ment comme exp st ; s est de I'ordre de ~-I(E/ET - 1) ou ET est le champ seuil. Le fait que nos resul- tats soient obtenus analytiquement plut6t que par des techniques de simulation numerique nous permet de donner une interpretation physique a nos predictions.

Dans la seconde partie, nous presentons quelques expkriences que nous avons faites pour compa- rer

a

la theorie. Nous mesurons I'effet d'un champ exterieur stabilisant - qui diminue TH

-

sur les courbes de seuil et nous mettons en kvidence le quenching du regime conducteur dans un khan- tillon normal (epais), ce qui conduit B un regime dielectrique B frequence nulle : ceci permet une mesure de 5 2 . Dans l'experience contraire, nous observons l'effet de la decroissance de z due au

vieillissement conduisant

a

I'apparition du rCgime conducteur dans un Bchantillon mince. Dans une

autre serie d'experiences, nous regardons les effets pretransitionnels. Nous mesurons directement s

en mesurant la dkpendance temporelle de la lumikre diffusk par I'oscillation entretenue p r h du seuil ; une telle experience nous permet de mesurer z simplement et directement. Dans deux autres experiences, nous observons Pintensite ( C C s-1) et la fonction d'autocorr6lation de la Iumikre diffuske par les fluctuations thermiques prks du seuil. Les expbiences sont en accord satisfaisant avec la theorie. Nous presentons aussi quelques comparaisons avec les resultats experimentaux d'autres chercheurs.

Abstract. - We explore the problem of electrohydrodynamic instabilities in nematic liquid crystals in the parallel (planar) orientation, in the context of the one-dimensional time-dependent model proposed by Dubois-Violette, de Gennes and Parodi. In the first part of the paper, we solve the equations of motion analytically in the case of an applied electric field E whose time variation is a square wave of frequency v. Our solution covers the entire range of material parameters, includ-

ing explicitly the case of either positive or negative dielectric anisotropy, and is valid for

0 5 v

5

10 kHz for typical materials. One result is a determination of the instability threshold curve, showing in particular that the conduction-regime threshold is a smooth closed curve in the E - v plane whose shape depends on the dielectric relaxation time z, the viscoelastic relaxation time

TH, and the Helfrich gain parameter (2. We treat the case of any sample thickness d and show that for samples so thin that T H (GC d2) <: zl2/(c2 - 1) the conduction-regime threshold loop shrinks and disappears, the dielectric regime then extending down to v = 0. We also treat in a very simple analysis the case of non-linearities and saturation when the applied field is above threshold. We are thus able to account for the variable grating mode (which has been experimentally observed), in

which the domain width is inversely proportional to the applied field over a factor of at least ten in domain width. We calculate also the growth (or decay) rate s of an oscillation near threshold, whose slowly varying amplitude goes as exp st ; s is of order z-l(ElE~ - 1) where ET is the threshold field. The fact that our results are based on analytic rather than numerical simulation techniques allows us to give a physical interpretation to the predictions.

(*) Laboratoire associk au C. N. R. S.

C1-238 I. W. SMITH, Y. GALERNE, S. T. LAGERWALL, E. DUBOIS-VIOLETTE AND G. DURAND In the second part of the paper, we present some experiments we have done for comparison with the theory. We measure the effect of an external stabilizing field, which decreases TH, on the threshold curves and show the quenching of the conduction regime in a normal (thick) sample, leading to the zero-frequency dielectric regime : this allows a measure of l z . In the converse experi- ment, we observe the effect of decreasing z due to ageing leading to the appearance of the conduction regime in a thin sample. In another series of experiments, we look at pre-transitional effects. In one experiment we measure s directly by observing the time dependence of Iight scattered from a forced oscillation near threshold ; such an experiment allows one to measure z simply and directly. In two

other experiments, we observe the intensity (cc s-1) and autocorrelation function of light scattered by thermal fluctuations near threshold. The experiments are in satisfactory accord with theory. We also present some comparisons with experimental results of other workers.

1. Introduction. - EIectrohydrodynamic instabi- lities in liquid crystals have already been the subject of much interest, both theoretical and experimental. The vast majority of this work has concentrated on nematic liquid crystals with negative dielectric aniso- tropy, in experimental geometries such that the exter- nally applied fields are stabilizing (opposite case of the Freedericksz transition). Most of the theoretical work has been done on a model in which spatial variation occurs only along one dimension. The distortion wave vector is parallel to the initial director, which lies in the plane of the parallel-oriented sample

-

a capacitor of liquid crystal between transparent electrodes (cf. Fig. 1). The first threshold estimation for static

f electrodes l a )

Pa

0 E l b )

&#+qL/a-~

4

_z n -sl, t x

FIG. 1. - Sample geometry. (a) Undisturbed orientation. The electric field is applied between the two electrodes. (6) In the presence of the field E, the situation in (a) may be instable for certain combinations of anisotropies in the properties of the medium. The anisotropic part of the induced dipole moment, Pa, and the currents due to the anisotropy aa of the conductivity

are indicated in the case c,, i el, o,, > al. fields was made by Helfrich [I], following an idea suggested by Carr 121. This DC excitation description was extended to time-dependent phenomena by Dubois-Violette, de Gennes and Parodi (DGP) [3] who wrote down the basic equations for the problem and discussed their solutions in a general way. The model proved unexpectedly difficult to treat and analytical solutions could be given only in certain

limits, cf. also [4], [5]. However, their conclusions, some of which were based on numerical computer solutions, still gave very rich and interesting predic- tions : the existence of two frequency regimes of instability with quite different dependence of threshold voltage on frequency and sample thickness ; a bizarre, s-shaped threshold curve ; the spatial wave-vector at which the instability appears (showing that the periodicity of perturbation may or may not be of the same order of magnitude as the sample thickness), and the important influence of diffusion currents at high frequencies. All of these predictions have been verified, usually quantitatively, by experiments a t Orsay and .elsewhere [6-91. At the same time, certain puzzling features, discrepancies, and domains excluded from consideration were apparent or appeared as time passed : the lack of analytic expressions for the various quantities, due to the numerical method of solution ;

the behaviour of the system for the case of thickness smaller than a critical thickness ; the very exaggerated s-shape for the threshold curve recently observed in certain materials [lo] ; the electrically-controllable

domain mode observed by Greubel and Wolff [11] and

by Vistin [12] ; and the disagreement between the theoretical value (2 or 3 V) and the observed value (7 or 8 V) of the low-frequency threshold.

DYNAMICS OF ELECTROHYDRODYNAMIC INSTABILITIES I N NEMATIC LIQUID CRYSTALS Cl-239

of the paper, we devote an Appendix in each of the two regimes to the details of the calculations whose results are quoted in section 4.

All our results for A. C. excitation are. based on square-wave excitation. The more usual experimen- tally used fields are sinusoidal, but the theoretical treatment for this case is much more difficult. In DGP a rough approximation is used which gives analyti- cally the approximate shape of the threshold curve, and a more precise but less illuminating numerical solution on a computer, evaluated only for

C2

= 3, is also discussed. A somewhat better but still inadequate analytical solution was given elsewhere by one of the present authors [13].

2. Formulation of the problem. - The mechanism

leading to instabilities in negative-anisotropy

(ell

<

cL) nematics submitted to an electric field was essentially pointed out in references [I-31. It will be convenient to briefly summarize the main features of the interaction and some of the important established facts. The basic ideas are schematized in figures 1 and 2. Let us consider

FIG. 2.

-

Instability mechanism. (a) In the electrically neutral medium with an applied E field, charge separation will occur wherever the director curvature is non-zero. The indicated flow pattern (double arrows) resulting from the charge transport will create destabilizing torques on the director (single arrows). Simultaneously, (b) the horizontal component Ez of E, due to the charges, will act on the anisotropic part of the induced dipole moment Pa and add a destabilizing dielectric torque as shown.

a geometry with unperturbed director no of the mole- cular alignment along the x direction, with an electric field E along the z axis. We assume a small fluctuation

A

6n in the director, n = no -t 6n where 6n = zn, cos kx varies only in the x direction. Curvature elasticity and electrical-anisotropy forces both act to restore the initial alignment, cf. figure l b : the system could be described by the equation

where T is a decay time given by

1

₊

Xa

H

'

+

X,, k') (1)

with

Here K,, is the bend elastic constant and yB is the corresponding viscosity relevant for bend deformations. In terms of other viscosity coefficients defined in refe- rences [4] and [lo] it is given by

To is the zero-E viscoelastic relaxation rate,

E, = ell - cl (< 0 here)

is the dielectric anisotropy and

X,

the magnetic suscep- tibility anisotropy. The magnetic field H could be applied along x as assumed in (I), or along z, in which case it is destabilizing and we would have to replace X, by

-

xa.

For H larger than a certain critical field (which depends on other elastic constants), To is then

<

0, i. e. the molecular alignment spontaneously dis- torts (Freedericksz transition). This latter case has been extensively studied elsewhere and will not be further considered.

Clearly the above equation predicts no instability at all in the medium ; there must be another physical parameter in the' system which interacts with the director. This parameter is the space-charge per unit volume. Suppose the material is slightly conduc- ting -and has a space-charge density q. The electri- cal conductivity of the material being anisotropic (oI1

>

ol), the spatially varying alignment n will cause charge to bunch up wherever the curvature

\Ir

E dnldx is non-zero ; in fact, positive charge will

build up wherever the curvature is convex

(-$I

E)

and negative charge where it is concave ($

//

E), cf. figure 2a. This charge will be acted on by E and will move, dragging the liquid along and creating a spa- tially-varying velocity field. The velocity v, as shown in the figure, will act on the alignment by exerting a torque on the director, which is proportional to curl v. Furthermore, the additional field 6E in the x-direction due to the space-charge distribution will add a dielectric-anisotropy torque of the same sign (Fig. 2b). Thus, the torques produced in this way tend to reinforce the original distortion n,, and we antici- pate that above a certain threshold the system may become unstable.

Let us write the complete equations for the time behavior of the amplitude of a given spatial Fourier

C1-240 I. W. SMITH, Y. GALERNE, S. T. LAGERWALL, E. DUBOIS-VIOLETTE AND G. DURAND

where we have included in the equation for $ the term due to the additional torques, both viscous and electric (i. e. due to 6E

-

q), with a coupling coefficient l/q which is the inverse of an effective viscosity :

In the equation for q, we have included the source term from the Carr-Helfrich mechanism with coupling constant o,,

in addition to the normal loss term due to conduction, with characteristic relaxation time z

The two coupled first-order differential equations (2) describe the time evolution of space-charge and curva- ture for any applied electric field. Due to the fact that the director relaxation time T i n general depends on time (by virtue of its E-dependence), the system cannot be integrated analytically in the case of general E(t).

Important physical insight into the nature of the solutions can, however, be obtained in special cases such as that of square wave excitation ;o be dealt with in this paper. As the zero frequency limit is important for comparison and reference, we will first briefly consider the static case.

3.1. THE DC THRESHOLD. - Under certain condi-

tions a static distortion of the director and a constant

.

.

spacecharge can exist. Putting $ = q = 0 in eq. (2), we

obtain two homogeneous equations in $ and q. For a

non-zero static distortion to exist, then, the deter- minant must vanish, i. e.

where we have introduced the gain parameter from (2)

We emphasize that this is the condition for a static

distortion to exist. As we shall see below, higher fields correspond to a growing distortion, lower ones to a decaying distortion. This electric field is thus the thres- hold for instability. If we consider materials with gain

parameter

c2

only slightly greater than 1 , we see the threshold becomes large. In fact, for

c2

<

1 there can be no instability for materials with E,

c

0. We refer the reader to Appendix I for a discussion of the case E,

>

0.

We note that, from the expressions of o, and z given above, the gain parameter can also be written

2

"

( =-

- % E l Y Ell

showing its dependence only on ratios and not on absolute values.

Some practical consequences of this will be discussed in the experimental section. In the absence of a magnetic field, T o is given by

Thus, we might expect the instability to appear at k = 0, i. e. at very long wavelength in the x direction, because here the threshold E is very small. However,

as pointed out by Helfrich, the effects of finite sample thickness d limit the lowest k possible for the distor- tion to about zld. For longer wavelengths, the viscous drag of the transverse return flow and the shorting out of the space charges by the conducting electrodes both act to reduce gain and to raise the threshold to increasingly higher values. This can also clearly be seen in the computer-calculated curves from the two- dimensional model by Penz and Ford [15] and by Pikin [16], and also in a calculation by Dubois-Vio- lette [I71 for an analogous thermal-instability problem.

Thus, applying the Helfrich condition

k 2 kH ( M nld), i. e.

r,

>

r,

= K,, kk/q,

with equality here, we obtain 2

( E d ) =

v2

= 4 n3 Ell K33

- &a &l(c2 - 1)

(5)

i. e. a voltage threshold, independent of sample

thickness. A formula equivalent to (5) was given for the first time by Helfrich [I].

We emphasize that, experimentally, D. C. excitation is difficult due to the presence of charge injection and charge depletion effects which invalidate the theore- tical treatment.

3.2. SIZE EFFECTS : RENORMALIZATION OF PHYSICAL

CONSTANTS.

-

When the thickness of the sample is of the same order of magnitude as the periodicity of the distortion along the x direction, the two-dimensio- nality of the problem becomes important. The distor- tion can no longer be regarded as pure bend, but has a non-negligible splay component due to the variation of

t,h

in the z direction. The presence of walls would cause an elastic relaxation of

t,h

even in the limit

DYNAMICS OF ELECTROHYDRODYNAMIC INSTABILITIES I N NEMATIC LIQUID CRYSTALS. CI-241 where we have introduced the splay elastic constant

K , , and the corresponding wave vector in the z direc- tion, k, x nld. In the context of eq. (I), this corres- ponds to a renormalization of the elastic constant

K3 3 + K3 3

+

Kl l(kzlkx)2

In fact, to correctly calculate the threshold when k, is not small compared to k, requires a full solution of the boundary-value problem. One of the authors has extended her treatment [17] of the formally similar heat convection (BCnard) instability in a nematic to the present case, and has found an expression for the threshold which is analogous to eq. (3), but in which all the physical parameters are replaced by effective values which depend on k,]k,. The renormalized value of

c2

(due to replacement of qB, o, E , etc. by their

renormalized values) for a material like MBBA

(5'

= 3) turns out to be 1.5 for the value of k,/k, which minimizes the threshold voltage. The renor- malized value of the DC threshold agrees quite well with the experimentally observed one, at about 8 V.

We thus expect the one-dimensional model to be quantitatively correct everywhere except where the wave-vector k, is of the order of or smaller than the inverse sample thickness. In these cases, renormalized values of the physical parameters must be used, though the unmodified model should retain a good qualitative validity even here. This means, in particular, that renormalization has to be taken into account throu- ghout the conduction regime.

3.3. NON LINEAR EFFECTS.

-

Up to the present, all treatments of electrohydrodynamic instabilities have been linear ones, able to predict thresholds and growth rates in the limit of small distortions. One interesting case where our one-dimensional model is able to make an important prediction is the problem of excitation above threshold. When a field higher than the threshold is applied, a real system will not exhibit exponential growth for all subsequent time but will saturate at a value of distortion determined by non-linearities. We will treat the problem at high fields in the context of DC excitation, which is a case that has been experimentally studied [11, 121.

Let us consider which important changes will take place in the basic equations (2) due to the presence of non-infinitesimal values of $ and q. We can see quali- tatively that the relaxation rate r , the viscous-drag

coupling 1/11, and the conduction-anisotropy pumping

efficiency o, will all be reduced at higher distortions. This will lead to a corresponding decrease in the gain

parameter

1'.

We expect this decrease to depend

directly on the deflection y of the director from the undisturbed direction, rather than on $, the first derivative of rp. In fact, one might expect the decrease of, for example, a, to be approximated by setting

a, -+ 0, c o s y

.

In any case, one expects to find

l2

+ C2(1

-

arp2)

where a is a number of order unity. Expressing this in terms of $ = k p (recalling that $ and p are Fourier amplitudes, not local values), we arrive at our principal assumption,.

which has the correct property of being independent of the sign of k ; we emphasize that this equation is valid when one spatial Fourier component $, is present. If we insert this in the DC threshold condition (3), we obtain

where we have emphasized that T o is also a function of the wave vector. This equation gives (implicitly) the saturation value of $ for a given excitation E, under the condition that only one Fourier spatial component of $ is excited. The situation Value $, given by eq. (6) would have some maximum at a specific k = k,, which could easily be found by solving eq. (6) for $

and taking the derivative with respect to k. We now suppose that non-linear interactions between different k-components will select out just the component with the highest saturation-dependent value, i. e. that $,

which corresponds to the largest value of the ratio of the k-dependent gain and the k-dependent saturation coefficient.

Performing the differentiation, we find the optimum wave-vector k given by the condition

This result is reassuring in that (for e. g. a 1) the saturation value of cp is reasonable, never being larger than about 450 for any value of

c2

> 1. Furthermore, we would expect the next term in the expansion of the saturation effects to be

-

y4, i. e. much smaller for

y2 =

3.

If we now insert this result into eq. (6), we obtain

Though this condition is similar in form to the DC threshold condition, it should be stressed that its physical significance is completely different : E is not some excitation threshold on the corresponding opti- mum k-value. Quite on the contrary, the E-value from (7) lies far above the threshold necessary for excitation of that particular wave-vector k.

Inserting the explicit expression for ro(k) we find, for k % n/d

C1-242 I. W. SMITH, Y. GALERNE, S. T. LAGERWALL, E. DUBOIS-VIOLETTE AND G. DURAND the optimum k for excitation increases linearly with

applied voltage. We emphasize that the values of k pre- dicted by eq. (8) lie well above the cut-off

k = kH x n/d for E

2

2 E,,

.

Thus, this result is independent of two-dimensionality renormalization, and offers (in particular) a direct measure of

c2.

Note that this effect should occur at DC in either the conduction or the zero-frequency dielectric regime (cf. sect. 4.3.3, d ) .

4.1. SQUARE-WAVE EXCITATION.

-

If the electric

field has the form of a square-wave of frequency v,

the difficulty of time-dependent coefficients in eq. (2) is circumvented and we may solve the equations exactly in any half-period ; 1/T remains constant since it depends on time only through ~ ' ( t ) . Taking the solutions for E = const of the form (in one half- period)

the characteristic equation of the system

yields the eigenvalues

a,,,

=

-

g(i

+

r ~ )

f

Ji(i

-

r ~ ) ,

+

c2(r2

-

r0

Z)

(9)

and the general solution

OH Ez

OH ET a , n ~ t / r

q(t) =

-

-- b eLtll'

.

(lob)

1

+

1, 1

+

A,

The behavior of the system thus depends on the ratios of dielectric to curvature relaxation times, z/T=rz,

and on the parameter

c2.

The solution (10) is valid as long as E is kept constant. It would also be a valid solution if we changed the sign of E. We remark that, as

changing the sign of E implies a corresponding change

in either q or $ but not both. That is, if q and $ is a solution for one half of the square wave period, during the other half period it will be either q and

-

$ or

-

q and $.

In the case of square-wave (or any other symme- trical) excitation, the system of eq. (2) admits exactly

two independent solutions (for any given k and

v)

quasi-periodic with period l/v, i. e. of the form

with f and g periodic of period l/v. It can be rigorously proved that the solutions can always be chosen of the form (ll), with s generally complex. In fact, for the problem considered here, the s will turn out to be real. In this case, we can choose, say,f, and g,, periodic in the half-period

3

v, and,f,, and g, antiperiodic, i. e. fI(t

+

+

v) = h(t> , hI(t

+

4

V) = - fII(t) (12)

g1(t

+

3

V) = - g1(t) 7 gII(t

+

t

v) = gII(t)

.

That this may be done follows from the fact that the equations are first-order in time and from the symmetry of E(t). We thus will impose on $ and q one of two conditions, which follow from eq. (11) and (12), and which correspond to two distinctly different physical situations :

Case I

This is, q changes sign twice each period l/v. This can happen only if space charges flow, and for this reason this regime is called the conduction regime.

Case I1

Here the space charge maintains the same sign always, and for vz $ 1, q is almost constant. Thus, by contrast to case I, this regime is appropriately called the dielec- tric regime.

4.2. THE NATURE OF CONDUCTIVE AND DIELECTRIC

INSTABILITIES.

-

If the only object of interest had been

the threshold condition we could at once have taken s

to be zero. As we shall see, however, an investigation of what happens slightly bellow and above threshold will enable us to grasp the physical processes determining the response behavior of the system. We will thus always work, in principle, with s # 0, specializing it to be zero whenever convenient.

Inserting the solution (10) into eq. (13) and setting t = 0 gives a requirement on A,, 1, for the conduction regime,

a

- s z

a2

-

ST

(1

+

1,) tanh = (1

+

1,) tanh ---

4 vz 4 vz (15)

DYNAMICS OF ELECTROHYDRODYNAMIC INSTABILITIES IN NEMATIC LIQUID CRYSTALS (21-243

+

(

exp 2

-

exp

U )

exp

s z ~ z A 2 zv sz

-

A, 1

-

exp

-

4 - -

-

2 vz _exp 1 2 _- t

+

EgH z - 1 - 1 , z sz

-

a

(exp

+

- exp

u)

2 vz -I- exp A1

-

t

.

- 1 - A 1 z Similarly, the requirement (14) together with the solu- tions (10) .gives for the dielectric regime

and sz

-

2 , -I- exp -

"-")

2 vz A , t

+

exp

-

.

1

+

A1 z

Eq. (15) and (17) give completely different situations for the possible choice of the eigenvalues A,, A, and their dependence on v, s, k and E. The discussion of this problem is found in Appendix I. For the moment, it is physically more instructive to use the characteristic equation to eliminate A, and A,, and set s = 0 (oscilla- tion at threshold). If we define the parameter A by

we then obtain, from (15) and (17), respectively,

+

L)

= (1

-

;)

sinh

&

(19)

zA sinh

(-

4 v T 4 v z for the conduction regime, and

zA zA sinh

(

1

+

L)

=

(IT

-

1) sinh

-

(20)

4 v T 4 v z 4 vz for the dielectric regime.

As we see from these equations the condition T = z

splits the problem naturally into two parts : for T > T there is no solution to (20) and consequently no

case v = 0), corresponding to an early observation by de Gennes [I%].

Let us begin the discussion with the conduction regime, T > z. For a normal sample, T & z in the absence of stabilizing fields and we already know of the existence of a low-field threshold at low frequencies. If we go to high frequencies, on the other hand, an expan- sion of the hyperbolic sines shows that an oscillation can exist only in the limit 1IT + 0, in contradiction to the fact that Tcannot exceed the value To (i. e.

r >

r,). Thus the instability vanishes at high frequencies, corresponding to the well-known experimental obser- vation that the conduction regime has a cut-off : the instability region is restricted to frequencies

5

1/z.

We have thus established the position of part of the conduction threshold curve. There will be another kind of threshold at fields so high that T

-

z. If we set T = z

in the expression for zA and take

for a normal sample (To p z), we obtain

Then, eq. (16) can be written

1

t'

2

1:

sinh - = 1

-

- sinh -

2 vz

(

2 vz

or, in the limit of low frequencies, vz 4 1,

Provided

t:

> 1, this relation can always be satisfied however close T comes to z, if only v is chosen small enough. Thus, in the limit of v = 0, this threshold occurs when E is high enough that T = z. For v # 0, it will occur at a value T w z, T being slightly higher, corresponding to a slightly lower field E. This means that for

>

1, which is necessary for the existence of

dielectric regime ; for T < 7 there is no solution to (19)

C1-244 I. W. SMITH, Y. GALERNE, S:T. LAGERWALL, E. DUBOIS-VIOLETTE AND G. DURAND

low-field threshold, there will also exist a high field branch of the threshold curve. We see the critical field E will be characterized by T z z and will decrease slowly with increasing frequency v. Combined with the existence of the cut-off frequency, this seems to indicate that the conduction-regime is a closed area in the (E, v) plane, which is also borne out by further investigation below. It is immediately clear that, as far as material parameters are concerned, this area will be more important, that is, will go to higher E and v values, the higher the values of

5'

and o (- 117).

The given arguments lead to a form for the conduc- tion regime threshold curve which would be qualitati- vely similar to that shown in figure 3 : a low-frequency, low-voltage branch ; a low frequency, high-voltage branch ; and a vertical portion near vz

--

1 which we would expect to join the two branches.

A rather complete understanding of the physics can now be attained by a simple discussion of the dynamic behaviour of I) and q. Consider the form of the oscilla- tion corresponding to different E-values in the conduc- tion regime area shown by figure 4. The position rela- tive to the loop is indicated by the inset in each case. At the point (a) on the lower branch, T % z, the curvature relaxation time is long (low E field) compared with the space-charge relaxation time z. The time dependence of

I) and q over a full period of the exciting field is shown to the left : let E reverse sign at t = 0 when $ and q are both positive and increasing. Since neither q nor $

can change sign instantaneously, both source terms in the differential eq. (2) will change sign, meaning that both q and $ start decreasing. With the response time of q much smaller, q will win the race and pass through zero first. Now the source term in the equation has turned positive and I) will start increasing. As q has now adjusted its sign to the changing sign of the field it will continue to grow in amplitude.

At a point such as (b), at a higher E field but at the same frequency, the corresponding oscillation takes the same general form, but with two changes ; the decay during the lossy time at the beginning of each half- period of E is more important since T is smaller (recall 1/T = AE2

+

T o ) and $ thus responds faster than at (a) ; and, during the gain time, the system more than recuperates from its losses and arrives at a higher value of $ and q after each period, i. e. s > 0, until the amplitude is limited by non-linearities. This occurs because the source terms are larger than at (a), being proportional to E.

When we go to higher values of E (ignoring for the moment the dielectric threshold curve), we reach a point where the oscillation has the form shown in (c).

The initial rapid decay of both q and I) now virtually quenches the oscillation since T x z : the time taken for q to reverse sign is almost enough for $ also to reach zero. In spite of the high E, the subsequent growth rates q and $ will be small at the beginning due to the smallness of the source terms (e. g. a, E*). When q and I) begin to reach appreciable values, however, the

FIG. 4.

-

Time dependence of q, y and y2 in a conduction

regime oscillation over a full period of the exciting field. The insets indicate the position in tlie E-v-plane. (a) z -=g T, vz 4 1,

s = 0 ; 11

-

VZ, 12

<

0. This is the case which goes over to the DC case. As the frequency goes to zero, the initial rearrangement

or loss time

-

z in length remains the same, while the following exponential growth rate goes to zero. That is, when a DC field is turned on, any existing fluctuation of q and y rapidly rearranges itself (i. e. decomposes itself into a-highly-damped dielectric mode plus a static conduction mode) to have the correct relation between v and q to support a growth rate of zero. (b) s > 0,

non-steady state ; (c) vz

<

1 ; 11 w 1,12 z -3 ; (d) vz

5

1,

z z T/2. In (d) the loss and gain regions are indicated, correspond- ing to negative or positive values of the product qE. The turn- around of y follows the change of sign of q, with a certain phase

lag.

growth becomes more and more explosive. If we now let T -+ z by further increasing the electric field E, the source terms will effectively vanish simultaneously

with q and $, both of which will now decay exponen-

tially, towards zero without ever changing sign. In other words : by increasing the field E we can comple- tely quench the instability. We thus see the physical origin of the restabilization (upper) branch of the threshold curve : we have approached sufficiently close to the line T = z where neither conductive nor dielec- tric oscillations can devdop. We also clearly see one important difference in character between the restabili- zation and the ordinary (lower) threshold branch :

close to restabilization (T % z) the system is extre-

DYNAMICS OF ELECTROHYDRODYNAMIC INSTABILITIES. I N NEMATIC LIQUID CRYSTALS C1-245

will change the behavior from oscillatory to non oscillatory. Thus, the upper branch is much more

explosive in character than the lower one, small

changes in E corresponding to appreciable changes in

the growth rate s.

Finally, if we establish an electric field of amplitude and frequency corresponding to a point not far from the cut-off limit, the oscillation will have the form given in (d). We notice that, since the electric field now has a much higher frequency, the time scale of the figure is altered relative to the earlier ones. In the time scale of

Ilv,

the response of q is getting slower and slower as we move up the lower branch in direction of higher frequency. With increasing loss time an increas- ing threshold is necessary to maintain the oscillation. With increasing E, the response of $ becomes more rapid, and with T and z more similar, a higher field is in turn needed to make up for the increasing losses, until the necessary field E is rising vertically as a function of frequency and the oscillation cut-off.

A last point that ought to be made about the conduc- tion regime concerns the wave-vector of the instability distortion. As already discussed the lower threshold is obtained by assuming that it corresponds to the lowest possible value of k , consistent with the Helfrich condi- tion ; the threshold field to excite a distortion of larger k is higher. On the other hand, the upper (restabilization) threshold is given by T

x

z, which means that

K33 2 1 1

A E ~

+

-

k = - z

-

= const.

Y e T z

Thus the restabilization threshold field for an instabi- lity of larger k is lower than for k = k, while the normal (lower) threshold field is higher. This implies that the threshold curve for a larger k lies within that for k = k,. Since s

<

0 for a given k everywhere outside its threshold curve, only for k = k, is the system unstable at the k = k, threshold curve, smaller values of k being excluded by the Helfrich condition k

>

k H z

nld.

Thus, the threshold wave-vector is kH everywhere in the conduction regime, in this theory. Note, however, that since the exact wave-vector k, where the oscillation occurs depends on the size effect on g2, which may be somewhat frequency-dependent, it is possible that k varies somewhat along the threshold curve despite the above considerations. For points inside the conduction regime loop we would expect the k-value to increase due to the non-linear effects discuss- ed above, although observation of this phenomenon would usually only be possible for a restricted region close to the lower threshold, because of the onset of dynamic scattering.

As the threshold k-value depends on the sample thickness according to the Helfrich condition k = k,

-

nld,

if we make the sample thinner and thinner, each new conduction regime loop will lie inside of every earlier one, corresponding to a higher

k-value both when diminishing E from the upper

branch and increasing E from the lower. If we continue

to decrease the sample thickness the conduction regime loop will shrink until it finally vanishes for a certain thickness. This squeezing-out of the conduction regime will be considered in more detail below.

The behavior of the system in the dielectric regime is entirely different. The basic condition, T

<

7 , has to be

achieved by giving E and k, or both, high enough values to keep T small.

To find the dependence of the threshold fieldon fre- quency, eq. (9) and (17) could be used to give an impli- cit relation between

r

and

r,

(and thus between E~ and-k). The real threshold field is found by minimizing E2 as a function of k 2 in the resulting formula : the threshold is reached when the first spatial Fourier component becomes unstable. We thus will have a prediction for the threshold field as well as for the wave- number k at which the instability first appears. Refer- ring to figure 5 we can state the physical reason for the existence of an optimum wave-number. Too small a

(V

I

"normal"

----_

...

instability

dielectric

FIG. 5.

-

Dielectric regime. (a) Disappearance of conductive instabilities for very thin samples. (b), (c) Time dependence of q, tyand y 2 ( T d z) (b) low-frequency case, vz

<

1, for a normal

sample (normally unobservable) ; (c) High-frequency case in the range of validity of eq. (35), i. e. - 1 2

-

vz & 1, I I z 1. The

case of a thin sample at low frequencies is similar to that of the

Helfrich branch of the conduction-regime threshold, with q

replaced by y and vice versa (Fig. 4a). ( d ) Low-frequency case. Here 1 2

-

- I/TH in analogy with figure 4a where I 2

-

-

l/z.

C1-246 I. W. SMITH, Y. GALERNE, S. T. LAGERWALL, E. DUBOIS-VIOLETTE AND G. DURAND value of k2 shows the response rate

r

=

l / T of $,

resulting in a larger fraction of each half-period's being spent in the lossy condition which obtains just after E reverses sign, whereas too large a value of k2 increases the elastic restoring forces and thus reduces the amplitude of the overshoot of

II/

into the gain configura- tion, necessitating a larger h to maintain the pumping of space-charge.

The discussion just given also explains physically why

we find k = kH not only at the conduction regime low

threshold, but also at the restabilization branch : E is here so high with a corresponding small 1/T value that an optimization would require a negative value of k2. Thus, the oscillation occurs with k E 0. Although in

the dielectric regime optimization of the wave-vector is normally observed up to high frequencies, an analo- gous case with k = kH will be found even in this region when the conditions are such that a high. field is neces- sary to maintain the oscillation, cf. section 4.3.3, e). 4.3.1 Quantitative calculations. - In order to obtain the algebraic form of the threshold curve, we would need to find a pair of eigen-values A,, A,, simulta- neously satisfying eq. (15) (conduction regime) or (17) (dielectric regime) and the characteristic equation. This could be done by numerical iteration. We have chosen different methods depending on the problem at hand, for instance analytical solutions based on series expan- sions. Mainly, however, we have used a combination of analytical and graphical solution which, apart from its being capable of giving analytical expressions for thresholds and other quantities, is physically quite illuminating, and which confirms the qualitative predic- tions given above. The methods of solution are exa- mined more closely in the Appendices I and 11. We summarize the results of the analysis below.

4.3.2 Results for the conduction regime. -

a) Low-frequency thresholds. - The conduction-regime

threshold curve is a closed loop, as sketched in figure 3. The low-frequency (Hewrich) limit of the lower branch has a value of electric field E given by

where in the absence of stabilizing fields

thus this is a voltage threshold. The low-frequency limit of the upper (Restabilization) branch is given by

this is essentially a field threshold since nor- mally To z

<

1.

b) Cut-off frequency for thick samples. - The

highest-frequency point on the threshold curve, the cut-of frequency v,, is given, for samples that are not too thin (d

2

3 dmin), by

4 v, z tanh (4 v, z)-' FZ 1

-

1/c2

.

(24)

For

5'

= 3, the value appropriate for MBBA, this gives

For comparison, an exact solution of the equation of motion in this same, one-dimensional case, by direct numerical integration [4], gives o, z = 1.21. This confirms the intuitive idea that the cut-off should lie around o, FZ l/z, determined by the conductivity of

the material. It should be noted, however, that, as

c2

diverges for 8, + 0, higher values for the cut-off frequency should in principle be possible, correspond- ing to especially high

C2

values. On the other hand, the one-dimensionality of the model should here be correct- ed for by using renormalized 1;'-values (cf. sect. 3.2) which tends to diminish the cut-off frequency.

c) Squeezing-out of conductivity instabilities. -

The conduction regime does not exist in samples where the calculated value of EH is greater than the calculated

ER. That gives the condition

for the existence of the conduction regime. This means, in the absence of stabilizing fields (e. g. H), that the thickness must be greater than a certain minimum

where dm, is given by

The formula also shows that the conduction regime will vanish in very pure samples (low enough o-value). Finally, the conduction regime can be squeezed out by stabilizing external fields. Consider for instance the case of a magnetic field H. Remembering that the relaxation rate is now given by

and that the conduction regime vanishes when

ro

has

been made large enough to satisfy

we find the critical value to which H has to be raised in

order to quench out even the lowest-k instabilities

DYNAMICS OF ELECTROHYDRODYNAMIC INSTABILITIES IN NEMATIC LIQUID CRYSTALS C1-247 4.3.3 Results for the dielectric regime. - a) The

low-frequency threshold. - For low-frequencies, the problem can be solved analytically, giving

This expression is valid to one per cent error for

Specifically, when the frequency tends to zero, we find

b) The optimum wave-vector at low frequency. - If we solve for k, we get a rather strange dependence on frequency :

Thus, k starts out at some finite value, decreases slightly until some v at which it passes through a minimum, and then increases again. We may easily find, by taking the derivative of eq. (30) with respect to v, that this minimum occurs for

For the case of

c2

= 3 (MBBA), this gives v* z=0.025, i. e. v* N 5 HZ in a relatively pure sample (z

--

5 ms).

As the dependence of k on frequency is very weak, for all practical purposes, we can set k = k (v = 0) in the low frequency region. A special consideration is needed for

C2

<

2.35 (see paragraph e) below).

c) Comparison of dielectric and conductive thresholds.

- Let us compare the zero-frequency dielectric

threshold with the zero-frequency conduction regime fields. We recall that the two fields for the conduction regime are given by

2 TH 7

AEH z = - (low branch)

C 2 -

1

and

A E ~ z = 1

-

rH

z (high branch)

with the condition for the existence of the conduction regime being

Thus

and

EH

<

ED < ER

.

That is, the dielectric threshold curve always intersects the conduction threshold curve, when the latter exists.

We note that the low-frequency dielectric mode threshold is usually unobservable, at least as a region between an undisturbed and an oscillating orientation, since this part of the threshold curve lies within the instability region for the conduction mode. Similarly, the lowest-frequency part of the upper branch of the conduction threshold lies within the dielectric-insta- bility region. However, one can in practice continue measurements of the threshold curves somewhat into the region to the left of the crossover point because the two modes have different wave-vectors and are thus distinguishable.

d) Thin samples. - If we increase T H z, e. g. make the sample thinner, to the point that the conduction regime just disappears, i. e. (25) becomes an equality, then the three threshold fields become identical. F:lr- thermore, from the expression for k2, with v = 0, we see the zero-frequency dielectric wave-vector is then exactly that of the conduction regime. If we continue to increase

rH

z, the dielectric regime can no longer choose its optimum wave-vector and oscillates with k = k,, under exactly the same condition as for the conduction regime, in particular, a threshold in voltage, not in field. Thus, the low-frequency dielectric regime, in a sample so thin that the conduction regime has been squeezed out of existence, mimics very closely the threshold and wave-vector behavior of the conduction instability. The only simple way to show the true nature of the instability under these conditions is to observe directly the time behavior of the curvature $ ; for the dielectric mode, $ changes sign twice each period, whereas in the conduction mode it maintains the same

sign. We note that most methods of observation (e. g.

homodyne light scattering, microscope observation) are sensitive to $', not to $. A heterodyne light scatter- ing measurement is necessary to distinguish between

+

$ and -- $.

C1-248 I. W. SMITH, Y. GALERNE, S. T. LAGERWALL, E. DUBOIS-VIOLETTE AND G. DURAND In quantitative terms, we find that

r z

- = const = 1.606

...

4 vz

or

Tz = 6.42 vz = 1.02 wz

.

(33)

For the value of T 0 / r which gives the mechanical frac- tion of the decay rate r , we find

= 0.217 for

c2

= 3 . Correspondingly, we find for the threshold

A E ~ 7 = (15.115~) vz

.

For a material with

c2

= 3, such as MBBA, this gives

AE2/v = 5.03

.

(34)

The frequency dependence of the threshold is then the well-known

E cc

.

We may also express k2 from our knowledge of To z :

Thus, the wave-vector will also show the same frequency dependence

k 01'2

.

For materials with

c2

<

2.35, on the other hand, the high frequency wave-number k is just the minimum possible, z n/d, i. e. the same as in the conduction regime. For these materials the dielectric regime threshold then occurs at k = k, and the threshold is approximately characterized by

We see that as

c2

approaches 1 both the zero and high

frequency limits of the dielectric threshold increase. We thus except the dielectric and conduction curves to intersect further to the left the closer

i2

is to 1 ; thus for a material with, for example,

c2

--

1.2, the c<s )) form

of the total threshold curve is extremely pronounced. This effect should exist also for sinusoidal excitation (though with a smaller overlap, as is shown for MBBA by the numerical solution of the equations of motion by Dubois-Violette [4]). Thus, the extreme s-shaped threshold curve which has been observed in a large

-

1

e, ( material 1101 is not surprising, since

c2

decreases toward 1 for larger

1

.salell

1.

Note that the high-frequency electric field (E2, not AE2 z) at threshold is essentially independent of e, for

c2

>

2.35 (i. e. for e, small enough) since, from eq. (34), E2/v

-

1/c2 A = q/oH 7. On the other hand, for such

large values of 8, that

c2

< 2.35 the high-frequency

threshold field increases.

f) Growth rate s near threshold. - Near threshold the growth or decay time of the oscillation is governed by the build-up or dissipation of space-charge, which takes a time of the order of or longer than z. Specifi- cally, we find

where the constant, calculated and plotted as a function of frequency and

5'

in Appendix 11, is of the order of 1 in most cases. Specifically, at zero frequency it is

4

for all values of

c2,

while the high-frequency limit is 1 for

5

'

>

2.35 and somewhat smaller for

c2

<

2.35 ; for

c2

M 1, the constant becomes asymptotically equal to

c2

-

1 a t high frequencies.

5. Limitations of the model. -There are two

effects which limit the validity of our calculation in addition to the approximate nature of the accounting for boundary conditions. First of all, our equations are derived in the viscous limit of hydrodynamics : the inertial terms are set equal to zero. This is an appre- ciable error only in the limit of small k or high fre- quencies. The ratio of inertial to viscous forces is of the order of poly, k2 and we can state the limiting condi- tion as

R r po/yB k2

<

1

.

(37)

For the normal conduction regime, the inertial terms are certainly negligible, as only the charges q are oscillating whereas the director curvature $ is practi- cally stationary in time. Near the restabilization limit of the conduction regime, this is however no longer true, cf. figure 4c. Here the time scale of the motion in $

is of the order of l/z. With z = 5 ms, k = nld and

q = 0.1 we find from (37) on replacing o by l/z, a value for R of 0.02 corresponding to a sample thickness

d of 100 ym. Thus inertial effects can be neglected in the conduction regime at least up to 1 mm sample thick-

ness, even in the two cases where $ is strongly modu-

lated : near restabilization or near the cut-off frequency. In the dielectric regime, for

i2

> 2.35, k2 increases proportionally to the frequency up to the Debye limit (see below), so that in this case, R is independent of frequency. From the condition (35) with

K,, = 8 x (cgs) its value turns out to be 0.04.

Thus, for materials with

C2

> 2.35 the analysis is valid in the whole dielectric regime up to the Debye limit. For

c2

<

2.35, k is always equal to k, and from (37) the analysis can be expected to be invalidated from about

v

2

1 kHz for samples of 100 ym thickness or

v

2

100 kHz for 10 ym.

DYNAMICS OF ELECTROHYDRODYNAMIC INSTABILITIES IN NEMATIC LIQUID CRYSTALS C1-249 is an ionic diffusion constant

-

lo-' cgs units, eq. (35)

gives us the ratio

x (Dl, vzyB/K3,) M lo-' Vz for

c2

>

3

.

Thus we expect k2 to depart from its predicted values for v

2

10 kHz in the normal dielectric regime. The case of k

2

k,, has been treated for sinusoidal excitation (DGP) ; the result is that k -+ n/d at high frequen- cies [22]. Thus, for all cases the high-frequency limit of the dielectric regime threshold field corresponds to eq. (36). We therefore will essentially always run into the inertial-term problem at high frequencies

2

10 kHz. In most cases studied so far the limitations described in this paragraph are of minor importance. However, for some recent work on instabilities in a medium where a relaxation in E , , causes E, to change sign

(becoming negative) above a frequency in the 10 kHz range [lo] the model has to be applied with more caution than in reference [19].

6. Experimental study of instabilities. - The mate- rial used in this work is MBBA (nematic between 15 0C and 47 0C) obtained from Merck and used without further purification. Some relevant physical data a r e : E~~ = 4.6, E , = 5.1,i. e.

-

E, = 0.55, = 1.12 ;a1

2

5 x 1 0 - l l Q - l c m - ~ , 0, /al, w 0.73, y1 = 76.3 cp, y2 = - 78.7 cp, yo = 1 4 . 2 5 ~ ~ . The die- lectric constants are the conventional ones. The visco- sities are taken from measurements by Gahwiller [20]. The conductivity usually lies much higher than the limit stated, and is subject to appreciable variations. From these data, one obtains

and z = &11/4 7~0,~

<

5 ms. y1 and yo further give y, = 12.0 cp and y = 6.0 cp. Then the value of the

dimension less gain parameter

c2

= aH z/yA turns out

to be 5.8, not about 3 as has been previously esti- mated [6]. On the other hand, a smaller value of the conductivity anisotropy could give a value as low as 3. As mentioned in section 4, this value should be expected to decrease, due to renormalization, for wave-vectors near the lowest value kH permitted by the sample size.

We note that, for a given type of nematic the total conductivity a may vary from sample to sample and, for a given sample, with age, due to slight chemical deterioration and consequent increase in the free-ion density (especially for samples exposed to electric fields). A sample cell freshly filled with MBBA, which is rather unstable chemically, starts off with a certain population of various positive and negative ions of different mobilities, 'leading to some particular value of oll/a,

-

1.2-1.5. As the sample ages, some parti- cular distribution of ions will be continually added to the population and will eventually dominate. Then the ratio a,,/o, will stabilize though o, and oil will conti- nue to decrease. Thus, following the remarks in

section 3,

c2

will also stabilize. We therefore expect just after a cell is filled, an evolution of both

1'

and

i,

followed by a stabilizing of

c2

and continued decrease of 2 . From published data, the ratio ol/oll is not too

well known, but has an uncertainty of about 50

%

giving an uncertainty of about 100

%

in

c2

; for MBBA it is commonly assumed that 1.7 <

c2

<

3.5. In our theoretical discussion, we have used the conventio- nally accepted value

c2

= 3 with the corresponding renormalized value of 1.5 although no conclusions would have to be altered in case of a higher c2-value. We point out that

c2

is usually measured by fitting the shape of the conduction-regime threshold, and thus corresponds, in a first approximation, to a renor- malized value.

All measurements were taken at room temperature on samples between parallel transparent conducting- electrodes (d between 7 pm and 50 pm). The parallel alignment was obtained by the rubbing technique.

6.1 QUENCHING OF THE CONDUCTIVITY REGIME. -

a) Influence of conductivity and sample thickness. -

In samples which are very pure (i. e. of low enough conductivity) and thin, no conduction regime will appear. If one makes a sample only slightly thinner than dm, for its value of z (cf. eq. (26)) and then gradually increases the conductivity (e. g. by ageing), the decreasing z can bring dm, down below d and the conduction regime will appear. As z continues to decrease,

r,

z will decrease proportionally ; a glance at Appendix I shows that the maximum frequency at which this regime exists will then evolve to greater and greater values.

To test these ideas, very thin samples (- 7 pm) were prepared. We then measured the threshold curve under

- -

square wave excitation and found it quite smooth down to a frequency of 2 Hz, see figure 6. At these frequencies, there are various irregularities which we ascribe to non-uniform charge injection at the elec- trodes, since the time it takes for an ion to cross the

C1-250 I. W. SMITH, Y. GALERNE, S. T. LAGERWALL, E. DUBOIS-VIOLETTE AND G. DURAND sample is comparable to this frequency. Microscopic

observation confirms that the sample takes on a non- uniform appearance, with rows of Williams domains forming along imperfections of the electrodes, for

v

5

2 Hz.

We now allow the sample to age a day or two and re-measure the threshold. There is a pronounced dip at low frequencies, which we identi& with the nascent conduction regime. As the sample ages progressively, i. e. as z continues to diminish, the conduction regime pushes further and further outward in fre- quency. Using as-a local oscillator a piece of diffusively scattering plastic mounted on the upper cover glass of the sample, we have verified by heterodyne measu- rements of the light diffracted into the first order by the instability that, even at a point such as P, in figure 6 , at very low frequency, the threshold labelled

dielectric is indeed the dielectric instability.

The form of the heterodyne signal, captured

with a transient analyzer, is shown in figure 7. The form of the threshold curves in figure 6 is very smooth, with no s-shaped region or discontinuity in curvature. As is easy to check, this feature is well borne out by the one-dimensional calculation : for T,z .. just small .

enough that the conduction regime can exist the threshold curves transform into smooth ones, for

c2

not too small.

10

-

Local oscillator level-

1

!

where the bar denotes averaging over a few periods of the high frequency field. So since

we write

r

= A E ~

+

r:

Then eq. (3) for the DC threshold becomes

The first term on the RHS is just the normal DC thres-

hold SO if we plot the difference of the squares of _-

the corresponding voltages, V: and

v:~,

versus V; we should obtain a straight line of slope l/(r2 - I), passing through the origin.

Using the same sample geometry as.in the preceding section we have measured a family of threshold curves with the hf stabilizing field as a parameter ; the results are shown in figure 8. The curves are labelled with

FIG. 7. - The signal scattered by the oscillation at threshold. Heterodyne measurement, dielectric regime (point P of figure 6).

b) Quenching by external fields. - The value of T , z greatly affects the shape of the threshold curve. Without decreasing d, T , can be increased by external electric or magnetic fields. To influence the threshold at constant sample thickness, we work with two elec- tric fields : the low frequency square wave field E and a high frequency field E,, of variable magnitude but of constant frequency, 2 000 Hz. The threshold field for instability at 2 000 Hz is far higher than the fields we work with. We thus may legitimately average the equations of motion (2) over a few periods of this field. Then the only place where this field affects the equations is in the formula for the curvature relaxation rate

r,

which depends on the square (here the short- term mean square) field :

FREQUE NCY(Hz)

FIG. 8. - Quenching of the conduction regime by an external high-frequency electric field (constant sample thickness). The threshold curves are labelled with the RMS high-frequency

voltage.

-

(v&)'/~ measured with a true-RMS voltmeter. The figure clearly shows how the conduction regime threshold curve shrinks and finally disappears as the stabilizing field is increased. The extrapolated DC values shown in figure 8 have been transferred to figure 9 as described above. We see that indeed the points are moderately well described by

corresponding to

c2

= 3.5. This implies that the bulk (non-renormalized) value of

c2

is somewhat higher,

w 5, which agrees well with the value estimated above