Critical thickness of a hybrid aligned nematic liquid crystal cell

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Critical thickness of a hybrid aligned nematic liquid crystal cell

G. Barbero, R. Barberi

To cite this version:

G. Barbero, R. Barberi. Critical thickness of a hybrid aligned nematic liquid crystal cell. Journal de

Physique, 1983, 44 (5), pp.609-616. �10.1051/jphys:01983004405060900�. �jpa-00209638�

Critical thickness of a hybrid aligned ^nematic liquid crystal ^cell

G. Barbero (*) ^{and R. Barberi}

Unical Liquid Crystal Group, Department ^of Physics, ^Calabria University, ^{I-87036 Rende,} Italy (Reçu ^{le 5 octobre} 1982, révisé le 6 décembre 1982, accepté ^{le 13} janvier 1983)

Résumé. 2014 Nous avons analysé ^la déformation des cristaux liquides nématiques ^{dans une cellule avec} alignement planaire ^{sur une} paroi ^et homéotrope ^sur l’autre. Nous supposons que l’interaction superficielle ^{est du} type (w/2) ^sin2 (~ - 03A6) ^et que l’énergie d’ancrage homéotrope ^est plus grande que celle relative à l’ancrage planaire.

En l’absence du champ électrique, ^nous démontrons qu’il ^{existe une} épaisseur critique dc au-dessous de laquelle

le cristal liquide nématique n’est pas déformé. Nous discutons aussi la dépendance en dc ^du champ ^réduit E/Ec (où Ec ^{est un} champ de seuil d’instabilité pour transition de type ^de Freedericks). Ec dépend seulement des énergies d’ancrage Wi ^sur les parois ^{et du} cristal liquide.

La dernière analyse ^montre qu’il y ^a deux épaisseurs critiques dc(H) ^et dc(P) : pour d dc(H) la cellule est homéotrope,

et si d > dc(P) ^{elle est} planaire. ^{Enfin nous} démontrons que si les énergies d’ancrage homéotrope ^et planaire ^sont égales, la cellule hybride ^est réalisable seulement si E/Ec ^1.

Abstract.

We discuss the director deformation inside a hybrid aligned ^nematic (HAN) cell, assuming ^{that the}

surface interaction is of the (w/2) ^sin2 (~ 2014 03A6) kind, ^{and we} suppose that the anchoring energy for homeotropic alignment ^is greater ^{than the} planar ^one.

In the absence of an external electric field, ^{we show that a} critical thickness dc, ^{below which} the nematic liquid crystal (NLC) ^is undistorted, exists. Furthermore we investigate theoretically dc ^vs. the reduced electric field

(E/Ec, where Ec ^{is a threshold field} depending ^{upon the} anchoring energies ^{and the intrinsic constants of the} NLC).

The last analysis shows that there are two critical thicknesses dc(H) ^and dc(P): ^{for d} dc(H) ^the sample ^is homeotropic (HOM), ^{while for d} > dc(P) ^{it is} planar (PLAN). Finally ^we point ^out that if the anchoring energies ^{are the same}

on the upper and lower plates ^{of the} sample, ^the HAN-configuration ^is realizable only if E/Ec ^1.

Classification

Physics ^Abstracts

61.30G - 42.65J

42.1OQ

Introduction.

The HAN cell has been recently

considered by ^various authors from ^a pratical [1-4]

and fundamental [5, 6] point ^{of view. In} fact this cell

seems to be particularly suitable for creating ^coloured displays [1, 2] ^or optical devices that might ^{use self} focusing ^to get bistability [4]. Furthermore the hybrid alignment gives ^a volume flexo-electric polarization : by analysing the behaviour of the HAN cell submitted

to an electric field it is possible ^to determine the bulk flexo-electric constant [6].

M. M. Labes et al. [4] ^{and L.} Terminassian-Saraga

et al. [7] show that the NLC film subject ^to antagonist homeotropic ^and planar anchoring ^{on two} glass plates ^can be distorted only if the thickness d of the

sample ^is greater than the critical one dc, ^and give

an order of magnitude ^for dr. ^{In the} present paper

we will calculate in a rigorous way dc, ^{and will} analyse

the influence of the elastic anisotropy ^K = I - K 11 /K3 3

and of the electric field on dc.

1. HAN cell without external field

The sample geometry is shown in figure ^{1. A NLC is} placed ^between

two glass plates (at ^{Z = -} d/2 ^{and Z =} d/2) ^coated

to induce homeotropic ^and planar orientations [8]

respectively. In this condition the sample ^{has under-}

gone ^a mixed splay-bend distortion. We can obtain the director deformation of the NLC by ^{the conti-}

nuum theory, supposing ^{that the} anisotropic ^inter-

action between the liquid crystal and the substrate is not strong. ^{In this} paper the surface energy is modelled as (wi/2) ^sin2 (Qi

oi), where wi ^{is the} anchoring strength coefficient, tP¡ ^the easy axis and (Pi the director angle ^at the surface substrate [9, 10].

In the case of the HAN cell we have tP 1 ^{= 0 and} tP2 ⁼ n/2 ^on the lower and _upper walls respectively,

and we suppose wl > w2 [11].

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01983004405060900

610

Fig. ^1.

^Schematic representation ^{of a HAN} cell in the

case of finite anchoring energy. T, and T2 represent ^the tilt angles ^{at the} interfaces (in ^the strong anchoring hypo-

thesis they ^are given by (p, = 0 and CP2 ⁼ n/2).

1.1 ONE CONSTANT APPROXIMATION.

In the one

constant approximation (K11 ^{1 =} K33 ⁼ K, i.e. K ⁼ 0) by minimizing ^the total free _energy we obtain

with the boundary ^conditions

where Li ⁼ K/wi ^are the de Gennes-Kleman [12]

extrapolation distances, the Z-axis is normal to the substrate and ₉ is the tilt angle formed between the director n and the Z-axis.

The subscript ^{Z means} the derivative with respect to Z.

By solving equation ( 1.1 ) ^{we obtain :}

where the constants Q2 ⁼ Q(d/2) ^and 91 1 ⁼ Q(2013 d/2)

are determined by equations (1.2). ^{In the strong} anchoring hypothesis wi -> oo and hence _{cp 1} ⁼ 0,

Q2 ⁼ n/2; ^then 9 ⁼ (n/2 d) (Z ⁺ d/2) [3, 6]. ^When

the anchoring energies ^{are finite, but} L d, ^{it is} possible ^{to linearize the} boundary conditions (1.2)

and after trivial calculations ^we obtain :

Consequently, ^from equation ( 1. 3)

In these cases (i.e. wi -> ^oo or Li d) ^for any sample-

thickness d the tilt angle Q is linearly increasing ^with

Z : this implies ^{that a} critical thickness dc, below which the sample ^is undistorted, does _not ^exist.

In the opposite situation, ^i.e. Li ~ ^{d, the} boundary

conditions give, ^for example,

By solving this transcendent equation for 2 Q 1 we find that

where h is an integer ^and

With the aim of determining the critical thickness dc,

it is necessary ^to find the situation where _QZ -> 0,

e.g. 92 - 9i

Then we can restrict ourselves to the case in which qJ¡ ^E (0, n/2). Using ^this assumption ⁱⁿ equation (2 .1 )

we have h ⁼ 0 and h ⁼ 1 only.

Furthermore, by taking ^{into account} that 0, 1 ⁼ 0, ø2 ⁼ n/2 and w1 > w2, ^we get Q2 > qJt’ and hence sin _(P2 > sin _{Q 1.} By using equation (2. .1 ) the latter condition can be rewritten as

These relations are equivalent ^to

Since L, L2, ^we deduce that h ⁼ I is not accep- table.

In consequence ^we have

and hence

The function _Q2 ⁼ Q2(d) is monotonically increasing

with d : this implies ^that equation (2.3) ^has only

one solution.

From equations (2.2), (2. 3) ^{it is} possible ^{to deter-}

mine _Qi, and then, by ^means of ( 1. 3), ^{the tilt} angle Q.

The transcendent equation (2.3) ^{has a} non-trivial

solution only ^if

which implies d > dc(H) ⁼ L2 - [1. ^{Then the} equa- tions (2.2), (2.3) ^{have the} unique solution Qi; ^{= 0}

if d dc(H) : consequently ^the sample is undistorted.

For d > dc(H), Qi # 0, i.e. it is possible ^{to obtain the}

HAN cell. We point ^out that for d dcH) ^{the NLC} subject ^to conflicting boundary conditions destroys

the smaller anchoring energy, reacting ^{to the} increasing distortion ; thus the result for the total free _energy is

w2/2.

The previous ^result, dc(H) ⁼ L2 - Li, ^{is in} agree- ment with that given by ^{M. M. Labes} [4].

It is possible ^{to rewrite} equation (2.2) ^as

Now it is _necessary to test the stability of the solu- tion _Q2 ⁼ 0 (i.e. (Q ⁼ 0 : the sample is in the HOM-

configuration) ^{for d} d, .

In this case the total free energy is given by :

where

’Ps represents the nematic-substrate interaction.

From equations (2. 2) ^and (1. 3) ^we find that :

and hence, by using Q2 ^as parameter, ^we obtain, from (5)

Consequently

From these relations we deduce that _T2 ⁼ 0 is a stable

configuration only ^{if d} dc(H’), ^{while for d} > dc (H)

the total free energy corresponding ^to HOM-situation is a maximum (see appendix 1).

Finally ^we point ^out that if the surface interaction is of the parabolic form the critical thickness does not exist.

_ _

In fact if ’YS ⁼ (1/L1) Q2 1 ⁺ (1/£2) (7t/2 CfJ2)2 ^the boundary conditions (1.2) ^{become :}

Taking ^{into account} equation (1.3) ^we find that the constants qJi ^are given by (1.4) and the tilt angle by (1. 5) : ^{hence the} sample ^is always ^distorted

1. 2 GENERAL CASE.

In the general ^case (K 11 K33, ^{i.e. K #} 0) equations (1.1), (1.2) ^{become :}

where Li ⁼ K33/Wi ^{and C2 is an} integration constant,

fixed _by ^the boundary conditions (6.2). ^From (6.1), (6.2) ^{we obtain}

being

By solving the simultaneous equations (7) ^{we can}

deduce the unknowns Q 1, Q2, Q and C (see Figs. 2).

Figures ^{2 show the} behaviour of _Q2 and _{Q 1} vs.

the sample-thickness d ^{for some elastic} anisotropies.

From figures ^{2 we see that} if d >> d c (H) the HAN cell

assumes a configuration ^near the ideal _one, with strong anchoring energies.

In order to determine dc(H) , ^{from (7.1)} written for Z ⁼ ± d/2 ^we determine, ^{after some} algebra,

(Pl (Pl((P2) ^and

From (8) ^{it is} easy ^to show that ðd( ({J2)la({J2 > ^{0 still,}

for _{any Q2} E (0, nI2), ^{and hence} dIHI ^{= lim} d(qJ2).

for _{any T2 c}

CP2 -+ 0

In the case of small _Q2, routine calculations give

Therefore, ^from (8), ^{we have} d((P2) ⁼ L2 - L1 ⁺ 0(Q2) ^{and then} dc,(H’) ⁼ L2 - L1, ^{as in the case of}

elastic isotropy; ^but Li ⁼ K33Iwi ^{if we} consider the situation L2 > L1(Wl > w2). ^{In the} opposite ^situation

an analogous calculation (where Q1 - n/2 if d - dc(p))

shows that the sample ^is planar ^{if d} d,(P) ⁼ L1 L2,

where Li ⁼ K 11 /wi. ^{In order to} examine the stability

of the HOM-configuration ^{for d} dc (H) ^{we proceed}

as in the previous ^case.

Now the total free _{energy W} is still given by equa-

612

Fig. ^2.

a) ^Plot of Q2 ^and b) of l{JI ^{vs. the} sample thickness d (J.1m), for K ⁼ 0, K ^{= 0.3 and K = 0.6, which are reasonable}

values [ 12]. ^{We note that} K > 1 only ^{next to the transition}

N - SA, ^and in this range the HAN cell is ^not used. The

case L1 ⁼ 1 J.1m and L2 ⁼ 5 J.1m is considered, giving dcH) ⁼ 4 um. ^We point ^out that lim CP2 ⁼ 7r/2 ^and

d-+ 00

lim _{Q1 1} ⁼ 0.

d-+ 00

tion (5), ^but f(Q, cpz) = ( 1 - ^{K sin2} Q) cpi. Using (6. .1 ) ^and (7 . 2) ^{it is} possible ^{to rewrite as}

In the limit of a small deformation close to the

HOM-configuration (e.g. rp2 ^- 0) ^we have from the

previous ^discussion

as was the case when K11 = K33. Hence the HOM-

configuration is stable for d dv(H).

We note that the present ^case, in the absence of an

external field, ^is very important ^{for the} analysis ^{of the}

non-linear birefringence ^{in a HAN cell} [5]. ^{Here for}

E Et, E, being the threshold field for the Freedericks transition, ^{it is} possible ^to expand ^{the tilt} angle ^{(p as a}

power series in e ⁼ E/Et. ^This way the zero approxi-

mation gives equation (7), ^while higher ^orders give

linear equations integrable only ^{if the zero order is}

known [14] (see appendix 2).

2. HAN cell submitted to external field.

Now let

us analyse ⁱⁿ greater depth ^{the case of a HAN cell}

submitted to an electric field parallel ^to the X-axis.

For a small positive dielectric anisotropy, ea ⁼ s ||

-

El [15], ^instead of (6 .1 ) ^{we have}

where ç -2 ⁼ (sa/4 nK33) E’, while the boundary

conditions remain as given by (6.2) [16]. ^The para-

meter ç ^{is the} well-known electric coherence length.

By proceeding ^{as in the} previous ^{case we obtain}

being

Equations (10) ^{solve the} present problem.

In this case the analysis ^{of the} existence of the HAN cell vs. the sample-thickness d ^{is more} complicated

than in the absence of an external field [ 17J. ^{In fact if d}

decreases the NLC tends to assume the HOM-confi-

guration, ^{in order to reduce} the total free _energy related to the elastic distortion, ^{as seen} above. But this

configuration may be unstable if the electric field is strong enough. Then in this situation, ^{the PLAN-} configuration ^{is more favorable} energetically.

In consequence ^{we must} consider the limit p2 -> 0+

(HOM-configuration) and CPt -> (n/2)- (PLAN-confi- guration).

2.1 TRANSITION HAN -+ HOM CONFIGURATION.

with reference to the case Q2 -> 0+ we deduce from

( 10 .1 ) : wi 1 ⁼ CPt (CP2). By substituting this result into

( 10 . 2) ^{we obtain d =} d(CP2). ^{For small} CP2 ^we have

Consequently, ^from equation ( 10 . 2), the critical thickness d,,, (H) ^{is given} by

From (12.1) ^we deduce that if ç ^is very small, e.g. the field is _very large (see below), d,(H ) ^{is given} by

while if -1 L 1 we ^obtain dr(H) ~ L2 - L1 - (1/2) ç - 2(L3 - L 3). As in section 1 for d dcH) ^{the sample}

assumes the HOM-configuration, ^{while for d} > dc(H) it is in the HAN-configuration.

It is _easy to show that if d dc (H) ^the HOM-configuration is stable. In this case the free energy density f(cp, cpz) appearing ⁱⁿ (5) ^{is :}

Taking ^{into account} equation (9) ^{we have :} f (Q, qJz) ^{= C 2 -} 2 ç -2 ^sin2 qJ.

Furthermore if we consider a small distortion near the HOM-configuration (i.e. qJ2 -> 0+) ^{we obtain}

Now, by observing ^that equations (11) ^and (9) imply ^that

and by putting dc,(H’)/d ^{= 1 + d, we} obtain the following form for the total free energy :

If 16 is very small [ 18], by expanding the square ^{roots as} power series in 6 we obtain, ^{after a} simple calculation,

in our hypothesis ^hence

2.2 TRANSITION HAN - PLAN CONFIGURATION. - Finally ^we examine the situation where _{w 1} -> (n/2) -.

By putting (pi ⁼ (n/2) - bi ^{we can} consider the ^case 6, -> 0+.

From ( 10 .1 ) ^{in this limit we have}

Obviously 62 is real and furthermore 62 ð1, ^{since lp2 = n/2 (i.e.} 62 ⁼ 0) is favoured energetically. Hence, ^{in our}

case where L1 L2, from (14) ^we conclude that the electric coherence length 03BE ^must be smaller than L 1 :03BE , L 1.

This implies ^{that the} HAN-configuration ^can degenerate ^{into a} PLAN-configuration only ^{if the} applied ^{field is} larger ^{than a critical one,} given by

A similar analysis ^{is not} necessary for equation ( 11.1 ), because (p 1 is real and furthermore (p 1 (P 2 for _any

]ç, ^{in our} hypothesis. ^In consequence the HAN-configuration may degenerate ^{into the} HOM-configuration ^for

any field, ^as may be expected.

614

The critical field E. ^{is needed to break the} anchoring ^on the non-favoured plate. If K 10 ~ " N, ga - ^0.1 [ 12]

and L2 ~ 5 J.1m, L 1 ~- 1 J.1m [ 19, 20] ^we deduce that E, n~ 107 V/m. Hence, by supposing L 1, ^{we find that}

From (16) ^and ( 10 . 2), by putting dc,( :’) ^{= lim} d((p ) ^we deduce that

ð1 -+ 0 +

It is _easy to show that if d dcP) ^the sample is in the HAN-configuration. ^{In fact,} the second term 0(d 1 ) ⁱⁿ equation (16) is of the form :

which, ^{from the} assumptions ç L1 ^and L1 L2, is smaller than zero. In consequence from (10.2), ^{we find}

that d dc(P) ^if d1 = ^{0. We} point ^{out that} ð1 # ⁰ implies 62 =A ^{0, from} equation (14), ^{i.e. the} sample ^{is in the} HAN-configuration.

From (17) ^{we note that} if ç --> Lt, dc --> oo. Further-

more if ç ^is very small with respect ^to L, (i.e. E > Ee), dcP) ^{is still} given by ( 12 . 2).

A calculation similar to that performed ^{for the}

HAN - HOM transition, shows that the PLAN-

configuration ^is stable if d > dc(p).

From the previous discussion it is possible ^{to con-}

clude that only ^{if the} sample-thickness d ^{is in the} range

dc, (H) _ dr (p) ^the HAN-configuration ^is realizable, ^while

in the cases d dc (H) ^{or d} > dc(,P) ^the sample ^can only

assume the HOM- or PLAN-configuration, respecti- vely.

Figure ^{3 shows} the behaviour of dc (H) ^and dcP) ^{vs. the}

reduced applied ^field (ç -1 = L-’(EIE,)).

From it we see that :

1. The HAN cell _may be realized only ^{in the shaded} region.

Fig. ^3.

Variation of the critical thickness dc(H) (lower line)

and dcP) (upper curve) ^{vs. the reduced} applied ^field E,-1 = L1(E/Ec), ^{in the case} L2 =5 um, L1 ^=lJ.1m.

The HAN-configuration ^{is stable} only in the shaded region.

2. If E/Ec ^{1 the} only limitation for the realiza-

bility ^{of the} HAN-configuration ^is represented by

d > dc(H).

In this region ^if L22> L21) the critical thickness is

approximatively given by dcH) ~ (L1I2)[2 - J2 -

3. If EIEc > > ^{1 the} range dc,(p) - dc(H) ^{is very small,}

then it is difficult to realize the HAN cell. We point

out that in this region dc(p) ^and dC(H) drop ^{to zero with}

the same behaviour.

4. If Li ⁼ L2, it is found that dc (H) ^{= 0 for} any electric field, ^while dC(P) undergoes ^an abrupt change ^if

the strength of the external field exceeds the threshold defined by equation (13) (see Fig. 4).

Fig. ^4.

^{In the case of} Li = L2 ^the HAN-configuration

is stable for any d only ^if EIE, 1, ^{while for} EIE, > 1 it is

always ^unstable.

3. Conclusion.

This work constitutes the first theoretical examination of the critical thickness in a

rigorous way. We have shown that in the absence of an

external field dc depends only ^on the bend elastic

constant (if w, > w2). Our results concerning ^the

dependence ^of dc ^{on the} anchoring energies wi ^{are in} qualitative agreement with those of M. M. Labes et al.

However, they ^{did not} consider the general ^{case where}

K =A 0 and E # ^0. In these situations we have shown that only ^{if the} sample-thickness d is in the _range

d , (H) - dC,(p) ^the HAN-configuration ^is stable, ^where

dC (H) ^and dC(P) ^are the critical thicknesses for the transi- tions HAN HOM and HAN PLAN respectively.

In particular ^we have demonstrated that the transition HAN - PLAN has a well defined threshold, EC, = (2/L,)VF7rK 33/Ea below which it is forbidden.

Obviously, ^{in our case,} the transition HAN HOM is always possible.

Acknowledgments.

One of the authors (G.B.) ^is grateful ^{to M. M. Labes for} suggesting ^this problem, ^to

G. Durand for supplying ^the preprint ^{of his} paper and to G.N.S.M. of Italian National Research Council that has partially supported this work. We wish to

thank C. Oldano for useful discussions.

Appendix ^1.

^{The case} L1 ⁼ L2 ^{must be} analysed

in separate ways.

In this situation from equations ( 1. 2) ^{we find that} equation (1.1) ^{has the} following solutions :

ponding ^to homeotropic, planar ^and hybrid configu- rations, respectively.

In the last case Q2 is determined by ^{means of} equa- tions ( 1. 2) ^written for _i ⁼ 2, and ^{is given by the trans-}

cendental equation (Lt ⁼ L2 ⁼ L)

From such a relation we deduce that :

a) lfJ2 is monotonically increasing with d,

consequently

A simple analysis ^{shows that the} i) ^and ii) ^solutions

are not related to stable configurations, ^{while the} hybrid configuration minimizes the total free _energy tl’, and hence it is a stable configuration.

In fact, ^{in this} case, ^we have Ys ^{s =} (sin2 qJ ^{1 +}

cos2 Q2)/ L, and after routine calculations we obtain

that are equal ^{to zero for the} boundary ^conditions ( 1. 2), ^and

Hessian determinant of the function Now it is _easy to show that

being Q2 ^E (nI4, nI2). ^{From the} previous relations, ^it follows that neither for _{cp = Q2} ⁼ 0 nor cp 1 = Q2 ⁼ n/2

the function T has an extremum, while for _{Q 1} ⁼

(n/2) - Q2 it has a minimum, ^since 02 tp 10lfJî ⁼ (21d) ⁺ (2/L) ^{cos 2 cp I} > 0, being Q1 1 E (0, n/4). ^We

observe that in this case dc,(H) ^{= 0, i.e. for} any sample-

thickness d we have QZ = ^{0 : the} sample ^is distorted, ^as

may be expected. By using this method it is possible ^to deduce dc ^{in another} way.

In fact in the general ^case (L1, = L2) ^{we obtain}

that are equal ^{to zero for} equations ( 1. 2), ^and

By noting ^that

being L2 > L1, ^{we deduce :}

is a stable configuration ^if

is not a stable configuration.

While if d > dc the stable configuration is distorted.

The method followed in the ^text gives ^{also some}

information about the behaviour of Q1 ^{1 and} Q2 ^vs. the

sample-thickness : ^{for this reason we have} preferred ^it.

616

Appendix ^2. - By putting ^and

into (8) ^we obtain, ^{for the zero} order in e, equation (6) ^for _Q(O) ^and Co. ^For the first and second order we have for Q(1) ^and Q(2) ^{two linear} differential equations, ^{that are} easily integrable. ^After simple calculations we get

where g(Z) ^{= 1}

^K sin2Q(o)(Z) ^{and yl, C12 are}

integration constants, ^fixed by ^the boundary ^condi-

tions (5.2). ^It is easy ^to show that C2 = y 1 = 0 and hence Q(1) ^{= 0. An} analogous procedure gives

The integration ^{constants are} now # 0. From

equation (5.2) ^we determine these constants, ^and

hence (P(2)’ ^{In this way it is} possible ^{to obtain the}

dielectric constant 82 [5], linear in the intensity, ^{with the}

relation :

where E ⁼ (E ^{+ e1.)/2, without solving the general}

problem.

References

[1] MATSUMOTO, S., KAWAMOTO, M., MIZUNAYA, K., J. Appl. Phys. ^{47 (1976) 3842.}

[2] RIVA, R., ^{Fis. Tecnol. 2} (1979) ^114.

[3] BARBERO, G., STRIGAZZI, A., ^{Fizika 13} (1981) ^85.

[4] HOCHBAUM, A., LABES, ^M. M., ^J. Appl. Phys. ⁵³ (1982)

2998.

[5] BARBERO, G., SIMONI, F., Appl. Phys. ^{Lett. 41} (1982)

504.

[6] ^DOZOV, I., MARTINOT-LAGARD, Ph., DURAND, G., preprint (1982).

[7] PEREZ, E., PROUST, ^J. E., TERMINASSIAN-SARAGA, L., MAUER, E., ^Colloid Polym. ^{Sci. 255} (1977) ^1003.

[8] Actually these conditions can be satisfied by putting,

as shown in ref. [7], ^a NLC between a free sur-

face (homeotropic orientation) ^{and water} (planar orientation).

[9] NAEMURA, S., ^J. Physique Colloq. ⁴⁰ (1979) C3-514;

WARENGHEM, M., ^Mol. Cryst. Liq. Cryst. ⁸⁹ (1982) ^15.

[10] COGNARD, J., ^Mol. Cryst. Liq. Cryst. Suppl. ^{Ser. 1} (1982) ^1.

[11] ^{In this} paper ^we suppose always that w1 > W2, i.e.

L1 L2. ^{In the} opposite ^{case our} calculation remain valid if we permute ~1 with ~2 and consider the limit ~1 ^~ (03C0/2) ^rather than ~2 ^~ 0. This situation is examined in ref. [4].

[12] ^{See for} example ^DE GENNES, ^P. G., ^The Physics of Liquid Crystals (Clarendon Press) ^1974.

[13] SCUDIERI, F., Opt. Commun. 37 (1981) 37 ;

SCHAAT, M., MULLER, F., IEEE Trans. Electron Devices 25 (1978) ^1125.

[14] BARBERO, G., STRIGAZZI, A., ^Mol. Cryst. Liq. Cryst.

Lett. 82 (1982) ^5.

[15] ^{DEULING, H. J.,} Solid State Phys., Suppl. ^{n° 14} (1979),

where the general ^case is considered.

[16] ^{In the case} of 03B5a 0 or in the case of 03B5a > 0 and E

parallel ^{to Z-axis, in} equation (9) the second term

becomes - 03BE-2 sin2 ~. ^This implies ^{that the}

calculations performed for 03B5a > 0 and E//X are

formally invariant if we replace the electric cohe-

rence length 03BE ^with i03BE, ^{where i2 = 2014 1.}

[17] ^{We are indebted to} the referees of ^« Le Journal de

Physique» for useful remarks on this section.

[18] ^The hypothesis |03B4| ~ ^{1 is not} necessary; it is per- formed in order to give ^a simplified ^{form to} equa- tion (13.2).

[19] NAEMURA, S., ^Mol. Cryst. Liq. Cryst. ⁶⁸ (1981) 183;

NAEMURA, S., ^L.C. alignment ^{mechanism and} influence

on display-device characteristics, preprint ^June

1982.

[20] LEVY, Y., RIVIÈRE, D., GUYON, E., ^J. Physique ^Lett.

40 (1979) ^L-215.